ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES

On the Rational Transformation between Two Spaces. 127

The following Paper, of which an account was given by tho Authorat the Meeting of tho Society on March 11th, 1869, was not receivedin time to take its place amongst the printed proceedings of thatMeeting (Vol. II., p. 174) :—

ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES.By PROF. CAYLEY.

Two figures are rationally transformable each into the other (or, say,there is a rational transformation between the two figures) when to avariable point of each of them there corresponds a single variable pointof the other. The figures may be either loci in a space, or locus in quoof any number of dimensions; or they may be such spaces themselves.Thus the figures may be each a line (or space of one dimension), eacha plane (or space of two dimensions), or each a space of three dimen-sions : these last are the cases intended to be considered in the presentMemoir, which is accordingly entitled, " On tho Rational Transfor-mation between Two Spaces." I observe in explanation (to fix thoideas, attending to the case of two planes), that any rational trans-formation between two planes gives rise to a rational transformationbetween curves in these planes respectively (one of these curves beingany curve whatever): but non constat, and it is not in fact the case,that every rational transformation between two plane curves thus arisesout of a rational transformation between two planes. The problem ofthe rational transformation between two planes (or generally betweentwo spaces) is thus a distinct problem from that of tho rational trans-formation between two piano curves (or loci in the two spaces re-spectively).

I consider,—in tho Memoir, (1) tho rational transformation betweentwo lines ; this is simply tho homographic transformation : (2) tho ra-tional transformation between two planes; and hero there is littloadded to what has been done by Prof. Cremona in his memoirs, " SulloTrasformazioni Geometriche dello Figure Pianc," (Mem. di Bolognat. II., 18G3, and t. V., 1865; see also " On tho Geometrical Transfor-mation of Plane Curves," British Assoc. Report, 186-1): (3) the ra-tional transformation between two spaces; in regard hereto I examiuothe general theory, but attend mainly to what I call the lineo-lincartransformation; viz., it is assumed that tho coordinates of a point intho one space, and tho coordinates of the corresponding point in theother space arc connected by three lineo-lincar equations (that is, eachequation is linear in the two sets of coordinates respectively). Tholineo-lincar transformation presents itself in the preceding two cases ;viz., between two lines, the homographic transformation (which, asalready mentioned, is the only rational transformation) is lineo-lincar;

128 Prof. Cayley on the '

and between two planes, the lineo-linear transformation is in fact the

well known inverse transformation (xl y'l z' = -'.-'.-). As regards\ • x y zf

two spaces, the lineo-linear transformation has not, I think, been dis-cussed in a general manner, and it gives rise to a theory of some com-plexity, and of great interest.

THE GENERAL PRINCIPLE OP THE RATIONAL TRANSFORMATION BETWEEN

Two SPACES.

1. In all that follows, the two spaces (lines, planes, or three dimen-sional spaces, as the case may be), or any corresponding loci in the twospaces respectively, are referred to as the first and second figures re-spectively. The two figures are in general considered, not as super-imposed or situate in a common space, but as existing, each indepen-dently of the other, as a separate locus in quo or figure in such locus.The unaccented coordinates (x, y"), (x, y, z), or (a?, y, z,w), as the casemay be, refer throughout to a point of the first figure; the accentedcoordinates refer in like manner to the corresponding point of thesecond figure.* Moreover X, Y,... are used to denote functions of thesame order, say w, of the coordinates (a>, y,...) : viz., (X, Y) are eachof them of the form ( # ]fcc,y)n; (X,Y,Z) each of the form ( # ]fo,ytz)n

t

(X, Y, Z,W) each of the form ( # ]£ x, y, z, w)n, as the case may be; andin like manner X', Y',... are used to denote functions of the same order,say n, of the coordinates (x, y\ . . .). This being so—

The condition of a rational transformation is that we have simul-taneously X : y... = X : Y ... ; as : y ... = X' : Y'...viz., these equations must be such that either set shall imply theother set.

2. If, to fix the ideas, we attend to the case of two planes, or take thesets to be x: y':z = X : Y : Z ; x : y : z = X': Y': Z',

• The coordinates (x, y) of a point in a line may bo concoived as proportional togiven multiples (a times, /3 times) of the distances of the poiDt from two fixed pointson the line; similarly the coordinates (x, y, z) of a point in a plane as proportionalto given multiples (o times, & times, y times) of the perpendicular distances of thopoint from three fixed lines in the plane; and the coordinates (x, y, z, w) of a pointin a space as proportional to given multiples (a times, 0 times, y times, 8 times) oftho perpendicular dibtances of the point from four fixed planes in the space. Obsorvothat even if the coordinates (x, y) and {sf, y'\ rofer to tho same line, and to tho sametwo fixed points in this line, they are not or necessity the same coordinates; viz., thefactors for x,y may bo a,/3, and thoso for sf^y1 may bo a',/3'. Tf those aro proportional(viz., if o : /3 = a' : /3'), then (x*, y1) will bo tho same coordinates of P ' that (x, y) areof P ; and in this case, but not otherwise, the equation xy'—xfy •>• 0 will imply thocoincidence of the points P, P' . The like remarks apply to the coordinates {x, y, z)

d ( )

Rational Transformation between Two Spates. 129

then' starting with the set x'.y'.z = X:Y: Z, for any given point(as, y, z) whatever in the first figure, we have a single correspondingpoint (as',y\ z) in the second figure; but for any given point (pa,y', z')in the second figure, we have primd facie a system of n1 points in thosecond figure, viz., these are the common points of intersection of thocurves x'.y'.z = X:Y:Z (in which equations x\y\ z are regarded asgiven parameters, x, y, z as current coordinates, and the equationstherefore represent curves of the order n in the first figure). Thecurves may however have only a single variable point of intersection;viz., this will be the case if each of the curves passes through the samen2—1 fixed points (points, that is, the positions of which are in-dependent of x'f y'_, z") ; and in order that the curves in question may eachpass through the v? — 1 points, it is necessary and sufficient that theseshall be common points of intersection of tho curves X=0 , Y=0, Z=0.[Observe that the condition thus imposed upon the curves X =Y=0, Z = 0 will in certain cases imply that the curves have ?ia com-mon intersections; or what is the same thing, that the functions X, Y, Zare connected by an identical equation, or syzygy, aX + /3Y + yZ = 0.This must not happen ; for if it did, not only there will be no variablepoint of intersection, and the transformation will on this account fail;but there would also arise a relation ux'+fiy'-\- yz' = 0 between(#', y\ z'), contrary to the hypothesis that (a/, y, z') are the coordinatesof any point whatever of the second figure. It thus becomes necessaryto show that there exist curves X=0 , Y=0, Z=0, satisfying thorequired condition of the w2—1 common intersections, but without aremaining common intersection, or, what is the same thing, withoutany syzygy aX+/3Y + yZ = 0.]

3. The curves x : y : z = X : Y : Z having then a single variablepoint of intersection, if we take (a;, y, z) to be the coordinates of thispoint, the ratios x : y -. z will be determined rationally ; that is, as aconsequence of the first set of equations, we obtain a second setx : y : z = X' : Y' : Z', where X', Y', Z' will be rational and integralfunctions of the same order, say ri, of the coordinates (a;', y', z') ; that is,we have a second set of equations, and consequently a rational trans-formation, as mentioned above.

4. It is easy to see that we have ri=n ; in fact, consider in the firstfigure a curve aX + /3Y + yZ = 0 , and an arbitrary line ax + by -f cz = 0 ;to these respectively correspond, in the second figure, the lineax' + fill' + yz' = 0, and the curve aX' + tY' + cZ '= 0 ; the curves areof the orders n, ri respectively, or the curve and line oi' the first figureintersect in n points, and the lino and curve of tho second figureintersect in ri points; which two systems of points must correspondpoint to point to each other; that is, we must have «'=«. It will

VOL. in.—No. 20. K

130 Prof. Cay ley on the

presently appear how different the analogous relation is in the trans-formation between two spaces.

5. Ascending to the case of two spaces, we have here the two sets

x':y: z : to' - X : Y : Z : W ; x : y : z : w = X' : Y ' : Z' : W,

the theory is analogous; the surfaces x' : y : z : vf = X : Y : Z : W(surfaces of the order n in the first figure) must have a single variablepoint of intersection, and they must therefore have a common fixedintersection equivalent to n3— 1 points of intersection: I say equivalentto n*—1 points, for this fixed intersection need not be ws — 1 points, butit may be or include a curve of intersection.* The surfaces X = 0 ,Y=0 , Z=0, W = 0 must consequently have a common intersectionequivalent to n8—1 points; there is (as in the preceding case) a causeof failure to be guarded against, viz., the condition as to the intersectionmust not be such as to imply one more point, of intersection, that is, toimply an identical equation or syzygy aX+/3Y+yZ + 3W=O betweenthe functions X, Y, Z, W; but it is assumed that they are not thusconnected. There is then a single variable point of intersection of thesurfaces x': y': z': vo = X : Y : Z : W ; or taking the coordinates ofthis point to be (», y, z, w), we have the ratios x : y : z : w rationallydetermined; that is, we have a second set of equations x : y : z : w =X ' : Y' : Z' : W, where X', Y', Z', W are rational and integral func-tions of the same order, say ri, in the coordinates (a?', y\ z't w') ; viz.,we have the rational transformation, as above, between the two spaces.

6. Suppose that the common intersection of the surfaces X = 0 ,Y=0, Z=0, W = 0 is or includes a curve of the order v ; and considerin the first figure the two surfaces

aX+/3Y+yZ + 2W = 0, ^ X + A Y + ^ Z + ^W = 0,

and the arbitrary plane ax + by + cz+dw = 0. The two surfacesintersect in the fixed curve v, and in a residual curve of the orderrt— v \ hence the two surfaces and the plane meet in v points on thefixed curve, and in n8— v other points. Corresponding to the surfacesand plane in the first figure, we havo in the second figure the twoplanes ax'+/3/ + yz + $w' = 0, atf+fty'+yxz + lyd — 0,and the surface ciX'+bY'+cZ'+dW' = 0 of the order ri: these inter-sect in ri points, being a system corresponding point to point with then*—v points of the first figure ; that is, we must have ri=n*—v. Andconversely, it follows that in tho second figure the common intersec-tion of the surfaces X'=0, Y'=0, Z'=0, W'=0 will be or includo a

• Tho curve of intersection may consist of distinct curves; each or any of whichmay bo a singular curve of any kind in regard to tho several surfaces.

Rational Transformation behvecn Two Spaces. 131

curve of the order v \ and that we shall have n^iP-rv Hence also

v—f'= (»—ri)

7. The principle of the rational transformation comes out more clearlyin the foregoing two cases than in the case of two lines, which from its'very simplicity fails to exhibit the principle so well; and I have accord*ingly postponed the consideration of i t : but the theory is similar tothat of the foregoing cases. We must have the two sets (each asingle equation) x : y = X : Y, and a>: y = X ' : Y'. The equationof : y = X : Y must give for the ratio x : y a single variable value;viz., there must be w—1 constant values (values, that is, independentof as', jy') ; this can only be the case by reason of the functions having acommon factor M of the order (n—1) ; but this being so, the commonfactor divides out, and the equation assumes the form a : y' = X : Y,.where X, Y are linear functions of (a?, y) : and we have then recipro-cally as : y = X' : Y', where X', Y' are linear functions of (as', y). Thusin the present case, instead of an infinity of transformations for dif-ferent values of n, n, we have only the well known homographictransformation wherein w=w'=l.

8. In the discussion of the "foregoing cases of the transformationbetween two planes and two spaces, it was tacitly assumed that n wasgreater than 1, and the transformations considered were thus differentfrom the homographic transformation; but it is hardly necessary toremark that the homographic transformation applies to these casesalso; viz., for two planes we may have x': y': z = X : Y : Z, andx : y : z = X': Y': Z', where (X, Y, Z), (X', Y\ Z') are linear functionsof the two sets of coordinates respectively ; and similarly for two spacesx': y : z : w' == X : Y : Z : W and x : y : z : w = X' : Y : Z' : W ,where (X, Y, Z, W), (X', T , Z', W ) are linear functions of the twosets of coordinates respectively. We may, if we please, separate offthe homographic transformation (as between two lines, planes, andspaces respectively), and restrict the notion of the rational transfor-mation to the higher or non-linear transformations; in this point ofview, the case of two lines would not be considered at all, but thetheory of the rational transformation would begin with the case of thetwo planes. Such severance of the theory is, however, somewhatarbitrary; and moreover the horaographic transformation betweentwo lines (being> a s mentioned, the only rational transformation) isanalogous not only to the homographic transformation between twoplanes, and to the homographic transformation between two spaces, butit is also analogous to the lineo-linear (or quadric) transformationbetween two planes^ and to the lineo-linear (which is a cubic) transfor-mation between two spaces.

K 2

132 Prof. Caylcy on the

9. For the sake of bringing out this analogy, I shall consider in somedetail the homographic transformation between two lines; but asregards the homographic transformations between two planes andbetween two spaces respectively (although there is room for a like dis-cussion) the theories may be considered as substantially known, and Ido not propose to go into them.

TnE HOMOGRAPIIIC TRANSFORMATION BETWEEN TWO LINES..

10. By what precedes, it appears that we have x : T/'= X : Y, where(X, T) are linear functions of (x, y) ; and conversely, x : y = X': Y',where X', Y' are linear functions of (a?', y) ; or what is the same thing,the relation is expressed by a single equation

(ax + by)x' + (cx+dy)y'= 0;or, as this may conveniently be written,

or, when the expression of the actual values of the coefficients is un-

necessary, ( # J»,y)(*',y') = 0.We thus see that the rational transformation between two lines is infact the homographic transformation; and also that it is the lineo-linear transformation.

11. A special caso is when ad— be = 0.

Writing here -5 = f = m, - K,a b a

the equation is (ax + by) (x'+my) = 0,that is (a,v + by) (a'x' + b'y') = 0;viz., either ax + by = 0, without any relation between x\y'; or elsoa'x'+b'y'= 0, without any relation between x,y; that is, to the singlepoint ax + by = 0 of the first figure there corresponds any point what-ever of the second figure; and to the single point a'x' + b'y'= 0 of thesecond figure there corresponds any point whatever of the first figure.

12. In the genei*al case where ad—be ^ 0, we may either by a lineartransformation (ax + by, cx + ih/ into y,—x or into a?,— y) of the coor-dinates of a point of the first figure, or by a linear transformation(ax' + cy\ bx' + dif into y\— x' or into x\—y) of the coordinates of apoint in the second figure (or in a variety of ways by simultaneouslinear transformations of the two sets of coordinates) transform the re-lation indifferently into either of the forms xy'—x'y = 0, xx'—yy'= 0;the former of these, or x : y-=. x : y, is the most simple expression of the

homographic transformation; the latter, or x': y'= - : - , is its exprcs-x y

sion as an inverse transformation.

Rational Transformation between Two Spaces. 133

13. If, to fix the precise signification of the coordinates (x, y), we employthe distances from a fixed point 0 in the link; taking the distances of thetwo fixed points (say A,B) to be a,j3, and that of the variable point P to bep, then we have x, y proportional to given multiples p (p—a), q (p—/3) of

the distances from the two fixed points; or writing •£. = n, we may say that

the coordinate - of the point P is = n t-—%; or in particular, if n = l , theny p-P

the coordinate is = P~~° If for w ^ 4 we write *~' ?„-, and thenP-/3 p-/3 p-ptake /3 = oo, we see that in a particular system of coordinates, A at O,

B at oo, the coordinate - is = p. Proceeding in the same manner iny t f .

regard to the coordinates (x\ y'), for a particular system of coordinates,.f

A' at O', B' at oo, the coordinate —7 of P7 will be = p . And the cor-y

respondence of the points P , P ' will be given by an equationapp'+bp'+cp+d = 0.

14. The equation just mentioned is often convenient for obtaining aprecise statement of theorems. %Thus taking A, B at pleasure on thefirst line, A', B ' the corresponding points on the second line, we have

p' = — ^ t r and thenceap + b

, ca-t-d /_ ,_(ad—bc)(p—a)a~ aa + b' . P ° ~ ( ~ 6 ) ( &)'

n,__cjl±d > ^ _P ~ p d' P ^ ~

J xi p—a aft + b p — aand consequently —,—~ = ———=• - -,

p—p aa + b p—pwhich is of the form ?.""", = m ^—^,

p —/> p —Pwhere (the correspondence app' + bp' + cp + d = 0 being given, and alsothe fixed points A, B) m has a determinate value not assumable a tpleasure. If, however, the fixed points A, B be not given, then we maydetermine a relation between them, such that m shall have any givenvalue not being = 1 ; we have in fact only to write

a/3 + 6 = ?n(aa + 6),that is a(/3 —ma)+J (1—m) = 0.( m = l would give a= /3 and the transformation would fail.) In par-ticular we may write m = — 1 , we have then


or the sum of the two distances OA, OB has a given value = de-ft

pendent on the transformation; one of these points being assumed atpleasure, the other is knownj the points A', B' are also known, andthe equation of correspondence is

it is moreover easy to show lhafc we have

15. In what precedes, the two lines are considered as distinct lines,not of necessity existing in a common space. But they may be con-sidered, not only as existing in the common space, but as superimposedthe one on the other. Suppose this is so, and moreover that the fixedpoints A', B' coincide with A, B respectively, and that the coordinates(a;, y) and (x, ij') are the same coordinates; so that the equationxy — x'y = 0 will imply the coincidence of the points P, P*.

16. If ad—be = 0, the equation of correspondence becomes(ax+hj)(a!x'+b'y') = 0,

and as before, to a single given point ax + by = 0, considered asbelonging to the first figure, there corresponds every point whateverof the line, or second figure: to a single given point a'as'+&'y'=O(the same as, or different from, the first point), considered as be-longing to the second figure, there corresponds every point whateverof the line, or first figure,

17. Excluding the foregoing case, or assuming ad—bcjtO, there arein general on the line two points such that to each of them consideredas belonging to either figure there corresponds the same point con-sidered as belonging to the other figure, or say there are two unitedpoints : in fact, writing as': y '= a?: y, we find aa5*+ (b + c) xy + dy* = 0, aquadric equation for the determination of the points in question. Un-less 4ad— (6+c)a = 0, this equation will have two unequal roots; andtaking the two points so determined for the fixed points A=A', B=B' ,the equation of correspondence will assume the form xy'—Jcx'y = 0.In this equation h cannot be = 1; for if it were so, the equation wouldbe xy'—x'y = 0 ; that is, the points P, V would be always one and thesame point. The equation may, however, be xy'-\-x'y = 0; the pointsP, P* are then harmonics in regard to the fixed points A, B. It is tobe observed, that if the equation xy'—lcx'y — 0 be unaltered by theinterchange of (as, y) and (as', y) we must have Jc1— 1 = 0, or since = 1is excluded, we must have k = — 1.

18. The original equation (ax+by)x'+(cx+dy)i/ = 0 is unalteredby the interchange only if b—c = 0; the equation 4ad—(6+c)*=0


becomes in this case ad—he = 0, which by hypothesis is not satisfied;the two distinct points A=A', B = B ' consequently exist. That is,if the correspondence between the two points P, F is such thatwhether P be considered as belonging to the first figure or to thesecond figure, there corresponds to it in the other figure the same pointP'—or say if the correspondence between the points P, P' is a sym-metrical correspondence—then as united points in the superimposedfigures we have the two distinct points A, B : and the correspondenceof tho points P, F is given by the condition that these are harmonics inregard to the points A, B.

19. There is still the case to be considered where 4<ad— (& + c)* = 0;the equation ax2 + (b + c)xy + dy1 = 0 has here equal roots, or the twounited points coincide together, or form a single point. Taking thispoint to be the point A, the coordinate whereof is x: y = 0 : 1, wemust, it is clear, have c?=0, and therefore also b + c = 0 : the relationbetween the coordinates (x, y) and (x, / ) is then axx +b (xy ~x'y)= 0 ; viz., this is the form assumed by the equation of correspondencewhen instead of two united points there is a double united point, andthis is taken to be the fixed point A.

20. It is to be observed, that we cannot have either 6 = 0, for thiswould give xx = 0, which belongs to the excluded case ad—be = 0 ;nor a = 0 , for this would give ccy'—x'y=z 0 : excluding these cases,the equation is of necessity altered by the interchange of (x, y) and(*'> y) ; that is, in the case of a double united point, the transformationis essentially unsymmetrical.

By what precedes, if the other fixed point be taken to be at infinity,the coordinates x : y and x : y' may be taken to be p, p respectively; viz.,p, p will denote the distances of the points P, P ' from the doubleunited point A; and the equation of correspondence then becomesf>p'+ b (p—p) — 0 ; that is, (p—b) (p' + b) +b2 = 0. .

21. The original equation axx'+ byx'+ cxy'+ dyy'= 0 can be reducedto the inverse form xx'—yy = 0 only (it is clear) in the symmetricalcase 6=c ; here, transforming to the united points, the equation is, bywhat precedes {ante, No. 17) xy'+xy = 0, which equation can be written(Ix+rny) (lx'+my') — (Jx—my) Qx'—my') = 0 , where I : m is arbi-trary ; viz., we have thus an equation of the required form.

22. In further explanation, start from the equation app' + b(p+p')+d= 0; that is, (ap + b)(ap'+b)+ad—b* = 0, or say (p—a)(p'—a)—tf= 0; this may be reduced to pp' —1 = 0; viz., the point O fromwhich are measured the distances p, p' is here the mid-point betweenthe two united points A, B ; and the unit of distance is | A B ; theequation expresses that the points P, P', harmonics in regard to the twopoints A, B, are the images ono of tho other in regard to the circle

136 . ' . Prof. Cayley on the

described upon AB as diameter. Take any two corresponding pointsL, L'; if the distances of these be X, X', we have XX'= 1; and hence

0>-X) (p'-X) = 1-X (p+P') + \2 = X(X+X'-p-p'),

i xi P—X o'— X X

and consequently £—— . — = — ; ^p—X p —X X

, . , ... x Jc(p~\) x Jc(p'which, writ ing - = — ^ — ^ —- = —*f

V P-* Vk2 = ~ (so that k* j± 1); or, k = X = I ,

X X

becomes xx'—yy'= 0; that is, the correspondence of the points P, F

being symmetrical, if the coordinate - of P be taken to be a multiple

of the ratio of the distances PL, PL' of P from any two corresponding

points L, L' (and of course the coordinate -y of P' to be the same mul-y

tiple of the ratio of the distances P'L, PL'), the equation of corres-pondence is obtained in the inverse form xx—yy '= 0.

THE RATIONAL TRANSFORMATION BETWEEN TWO PLANES.

23. Starting from the equations x': y '. z'= X : Y : Z, where X=0 ,Y=0, Z = 0 are'curves in the first plane, of the same, order n, it hasbeen seen that in order that we may thence have a rational transfor-mation between the two planes, the curves X=0, Y=0, Z = 0 musthave a common intersection of n2—1 points, and no more ; that is, theymust not have a complete common intersection of ri* points. In thecase «=2, taking the n2—1 points in the first plane to be any threepoints whatever, the condition that the curves shall be conies passingthrough the three points does not in anywise imply that the coniesshall have a common fourth point of intersection; and we have thus arational transformation as required; viz., the first set of equations isx':y: a ' = X : Y : Z, where X = 0, Y = 0, Z = 0 are conies passing.through the same three points of the first plane; and as it is easy tosee (but which will be subsequently shown more in detail), the secondset is the similar one x : y : z = X': T: Z', where X '= 0, Y'= 0, Z '= 0are conies passing through the same three points in the second plane;this may be called the quadric transformation between the two planes.

24. But the like theory would not apply to the case « = 3 ; if thena— 1 points in the first plane were any eight points whatever, thecubics X=0 , Y=0, Z = 0, intersecting in these eight points, wouldhave ar common ninth point of intersection, and the transformation


would fail; and so for any higher value of n, taking at pleasure any•£ra(r& + 3)— 1 of the n2—1 points of the first plane, the curves X = 0,Y=0, Z = 0 of the order n passing through these \n (n + 3) — 1 points,would have in common all their remaining points of intersection, andthe transformation would fail. A. transformation can only be obtainedby taking the n2—1 points in such wise that these can be made to bethe common intersection of the curves, and at the same time that thenumber of conditions imposed upon each of the curves X=0 , Y = 0 ,Z = 0 shall be at most = | n (n + 3)—1.

25. And this requirement may be satisfied; viz., the number of con-ditions may be made to be = \n (w+3) —1, by assuming that certainof the «2 —1 points of intersections are multiple intersections of thecurves. For if we have a given point which is an a-tuple point on eachof the curves X=0 , Y=0, Z=0, then this counts for a2 points of in-tersection of any two of the curves, and thus for a2 points of the ft2 — 1points : but the condition" that the given point shall be on anyone of the curves, say the curve X = 0, an a-tuple point, imposeson the curve, not a2, but only •£« (a + 1) conditions : and we have inthis way a reduction whereby the number of conditions for passingthrough the n* — 1 points can be lowered from w2—1 to the requirednumber £w(n+3) — 1.

26. In particular, for n=S, we may for the r?—1 points of the firstplane take a point as a double point on each of the cubic curves X = 0 ,Y=0, Z = 0 (which therefore reckons as four points), and take anyother four points. Each of the curves is determined by the conditionsof having a given point for double point, and of passing through fourother points; that is, by 3 + 4 = 7 conditions; and the three cubiccurves X=0 , Y=0, Z = 0 have for the common intersection the doublepoint reckoning as four points, and the given other four points; thatis, they have a common intersection of 4 + 4 = 8 points; but this doesnot imply that they have a common ninth point of intersection; wehave therefore a rational transformation as required; viz., the first setof equations is x : y': z = X : Y : Z ; where X=0 , Y=0, Z = 0 areeubics in the first plane having each of them a double point at thesame given point, and also each passing through the same four givenpoints ; the second set of equations is x : y : z = X': Y': Z', whereX '=0 , Y'=0, Z '=0 are like eubics in the second plane.

27. Generally suppose that the w2—1 points in the first plane aremade up of a, points, which are simple points; cr2 points, which aredouble points; a3 points, which are triple points, ... an_, points, whichare (n—l)tuple points (an_i = 1 or 0), on each of the three curves;these will represent a system of w2—1 points if only

138 Prof. Cayley on the

The number of conditions imposed on each of the curves X=0, Y=0,Z=0 will be ai + 3aa-}-6a3... +%n (n—l)an-\; for tho reason presentlyappearing, I exclude the case of this being <\n(n+3)—2; and there-fore assume it to be = %n (»+3) — 2. In fact, writing

a, + 3a, + 6a, ... +£M ( n - l ) o . . , = $n

this combined with the former equation gives

viz., the singularities are equivalent to £(n—l)(n—2) double points,that is, to the maximum number of double points of a curve of theorder n; or say each of the curves X=0, Y=0, Z=0 is a curve of theorder n having a deficiency = 0; that is, it is a unicursal curve of theorder n. Hence also, taking (a, b, c) any constant factors whatever,the curve aX+6Y+ cZ = 0 is unicursal.

28. It is important to remark that the conclusion follows directlyfrom the general notion of the rational transformation; in fact,the equation aX + 6Y + cZ = 0 is satisfied if as : y : z = X': Y': Z';az'+by'+cz' = 0. The last of these equations determines the ratiosx'ly: z in terms of a single parameter (e.g. the ratio x: y'), and wehave then x : y : z expressed as rational functions of this parameter;that is, the curve is unicursal.

29. Suppose for a moment that it was possible to havea,+3a,+6a,... +±n(n-l)^.! < | n (»+3) -2 .

Combining in the same way with the first equation, it would follow that

which would imply that the curves X=0, Y=0, Z=0 break up each ofthem into inferior curves: but more than this, the coefficients a,btc beingarbitrary, it would imply that the curve aX+6Y + cZ = 0 breaks upinto inferior curves; this can only be the case if X, Y, Z have a com-mon factor, say M; that is, if X, Y, Z = MX,, MY,, MZ,; but wecould then omit the common factor, and in place of x'l y'\ z = X : Y ; Zwrite x\ y: z — X, : Y, : Z,, where X,=0, Y,=0, Z,=0, are propercurves, not breaking up; the above supposition may therefore be ex-cluded from consideration.

30. We have thus a transformation in which the first set of equationsis as': y': z — X:Y:Z, where X=0, Y=0, Z=0 are curves in tho firstplane, of the same order n, having in common a,,a,... oB., points whichare simple points, double points,... (n—1)tuple points respectively oneach of the curves, these numbers satisfying the conditions


conditions which give, as above,aa+3a3 ... + * ( n - 1 ) ( n - 2 ) an^ = | (w-1) ( n - 2 ) ,

and also a1+2aa+3*8.. . +(w—1) aB_j = 3n,—3;so that the relations between ai, a3... an.} are given by any two ofthese four equations.

31. The second set of equations then is x : y : z = X': Y': Z', whereX '= 0, Y'= 0, Z '= 0 are carves in the second plane, of the same ordern; and it is clear that these must be carves such as those in the firstplane; viz., they must have in common a\, a2, ... a'n.i points, which aresimple points, double points, ... (n—1)tuple points respectively oneach of the curves, the relations between these numbers being expressedby any two of the four equations

«i+3aa+6a'3...aa+3a8... )

al + 2aa + 3aj,. . + (n-l)a'n_x = 3»—3.32. To any line aaf+by'+cz'=: 0 in the second plane there, cor-

responds in the first plane a curve aX-f-6Y+cZ = 0 of the order w;and to any line a'x+b'y + cz = 0 in the first plane there corresponds inthe second plane a curve a'X'+fc'Y'+c'Z' = 0 of the same order w; thecurves aX+JY+cZ = 0 in the first plane are, it is clear, a system, andthe entire system, of curves each satisfying the conditions which havebeen stated in regard to the individual curves X = 0, Y = 0, Z = 0, andbeing as already mentioned nnicursal; and similarly the curvesa'X'+iTT'+c'Z' = 0 in the second plane are a system, and the entiresystem, of curves each satisfying the conditions which have been statedin regard to the individual curves X '= 0, Y '= 0, Z '= 0; and being alsounicursal. We may say that to the lines of the. second plane therecorresponds in the first plane the reseau of curves aX+5Y+cZ = 0;and to the lines of the first plane there corresponds in the second planethe reseau of curves a'X'+6'Y'+c'Z' = 0; these reseau x being sys-tems satisfying respectively the conditions just referred to.

33. We have next to enquire what are the curves in the second planewhich correspond to the ax+Oj... -fa,,.! points of the first plane. Iremark that the <*i+aa... +oB_i points are termed by Cremona theprincipal points of the first plane, and the corresponding curves theprincipal curves of the second plane. But it will be more convenient tosay that the oi + oa ... +an.\ points are the principal system of the firstplane, and the corresponding curves the principal counter-system of thesecond plane. And of course the aj + aa... +a'w_1 points will be theprincipal system of the second plane, and the corresponding curves theprincipal counter-system of the first plane.


34 The Jacobian (curve) of the curves X = 0 , Y = 0, Z = 0 is, ofcourse, the Jacobian of any three curves aX+&Y+cZ = 0 of the firstplane, or it may be called the Jacobian of the reseau of the first plane;and similarly, the Jacobian of the curves X '= 0, Y'= 0, Z'= 0 is theJacobian of the reseau of the second plane.

35. I say that to each point at of the first figure there correspondsin the second figure a line; to each point a2 a conic; to each point a3

a nodal cubic; ... and generally, to each point ar a unicursal r-thiccurve; the entire system of the curves corresponding to the «i + aj+«j... + a».i points, that is, the principal counter-system of the secondplane, is thus made up of at lines, a2 conies, a3 nodal cubics, ... ar uni-cursal r-thics, an_i unicursal (ti— l)thics. It is thus a curve of theaggregate order a1 + 2ai+Saa... + (n—1) atl.u = 3n—3; and it is infact the Jacobian of the reseau of the second plane; as such, it passesthrough each point a[ two times, each point a'a five times, ... each point a'r3r—1 times, ... each point a'n_i Bn—4 times.

36. The reciprocal theorem is of course true. The Jacobian of thereseau of the first plane is thus made up of a\ lines, a'% conies, a, nodalcubics, ... a'r unicursal r-thics, ... a'n_x unicursal (?i—l)thics. Calcu-lating the Jacobian of the reseau of the first plane, we have thus thenumbers a'u a'it ... a ,.1} which determine the nature of the principalsystem of the second plane.

37. I indicate as follows the analytical proof of the theorem that toa principal point ar of the first plane there corresponds in the secondplane a unicursal r:thic. Consider the simplest case, r = 1 ; if in theequations x : y': z = X : Y : Z the coordinates (as, y, z) are consideredas belonging to a point au these values give identically X = 0, Y = 0,Z = 0; hence for the consecutive point x + &B, y + Sy, z + $z, if (A, B, C)denote the derived functions of X, (Au Blt Cj) those of Y, (A2, B2, Ca)those of Z, we have

x : y : z = A&e+B hj-\-C fa

We have A, B, C = 0, for the determinant is the value, at the

point <*! in question, of the Jacobian of the reseau of the first piano;and the Jacobian curve passing through c (in fact, having there adouble point), the value is = 0 .

38. Hence x', y\ z', considered as corresponding to a point indefinitelynear to oi, are connected by a linear equation. Corresponding to <xx we

Hational Transformation between Two Spaces. 141

have in the second figure a line. But it is to be observed, further,that the equation of the line is that obtained by writing in the fore-going equations say fa = 0, and eliminating the remaining quantitiesfix, Sy ; or what is the same thing, we may consider the equation of theline as given by the equations

. x : y : z = A &B+B hj

where §x, Sy are indeterminate parameters to be eliminated.

39. In the case of a point ar we have in like manner

x': y : z = (a, ...]£&B, Sy, fa)r

: (a,, ...£&e, Sy, fa)r

where (a, . . .) , (CTI, . . .) , (a^, ..'.) are the r-th derived functions ofX, Y, Z respectively. In virtue of the relation of the point ar to thecurves X = 0, Y = 0, Z = 0, the coefficients will be such as to allow ofthe simultaneous elimination from these equations of the three quan-tities £a;, Sy, fa. . The result of the elimination will be the same as if,writing say fa = 0, we eliminate &e, oy; or what is the same thing, therelation of as', y\ z' may be regarded as given by the equations

x' I y : z = (a, ...][&», 8y)r

; (a,, ...][&B, fy)r

: (02, . . . ;

where ?», $y are indeterminate parameters. These equations obviouslyexpress that the point (x, y't z) is situate on a unicursal curve of theorder r.

40. It is further to be shown that the r-thic curve thus correspond-ing to ar is part of the Jacobian of the reseau of the second plane. TheJacobian in question is the locus of the new double point of thosecurves of the reseau which have a new double point; that is, a doublepoint not included among the 02 + 03... +a'n_l singular points of theprincipal Bystem of the second plane. But a curve of the reseau beingunicursal, can only acquire a new double point by breaking up intoinferior curves. Consider, in the first figure, any line through ar, thecorresponding curve in the second figure is made up of the unicursalr-thic curve, which corresponds to the point ar, together with a residualcurve variable with the line through ar; this is a unicursal curve of theorder n—r. The aggregate curve of the order r + (?i—r) has singular


points equivalent to ^(w—1)(»—2) + l double points;* that is, thesingularities are those belonging to the principal system of the secondplane, together with a new double "point constituted by an intersectionof the curves r, n—r. [Observe that the two curves have only thissingle intersection; viz., the remaining r(n—r)—l intersections are atpoints a'3+a3... +a'n_x of the principal system of the second plane.]We have thus, in the second plane, a series of curves, each of themhaving a new double point; viz., these are the several curves whichcorrespond to the lines through ar in the first figure. Each of thecurves is a fixed curve r together with a variable curve n—r. Thenew double point is an intersection of the two curves; that is, it is avariable point on the curve r. The locus of the new double point isthus the curve r ; therefore the curve r is part of the Jacobian of thereseau of the second plane. Since each point ar gives a curve r,the curves in question form an aggregate curve of the ordera,-f2a + (n—l)aw_i, =3w — 3 ; viz., this is the order of theJacobian; or, as stated, the curves r (that is, the principal counter-system of the second plane) constitute the Jacobian of the reseau ofthis plane.

41. The numerical systems (ab oa.. . on.,) and (ai, a\... a'n.x) areeach of them a solution of the same two indeterminate equations

Sr1oP = na—1, 2 r a r = 3 » - 3 ,

but not every solution of these equations is admissible; for instance, ifr>\n, then ar is = 0 or 1, for ar = 2 would imply a curve of theorder n with two r-tuple points, and the line joining these would meetthe curve in more than r points ; similarly, r>-Jn, ar is = 4 at most, foror = 5 would imply a curve of the order n with five r-tuple points, andthe conic through these would meet the curve in more than 2n points;and there are of course other like restrictions. The different admissiblesystems up to n = 10 are tabulated in Cremona's Memoir; and he hasalso given systems belonging to certain specified forms of n: theseresults are as follows:—

• I n general, if r + r ' = «, and tho curves r , »•' arc each unicursal, then tho ag-gregate singulari ty arising from tho singularities of tho two curves and from theiri n t s e c t i o n s is e q i v a l e n t to * ( 1 ) ( 2 ) £ ( ' 1) (/ 2) y t h t i tgg g y g gintersections, is equivalent to * ( r - 1 ) (r— 2 )+ £(>• ' - 1) (/— 2) + n y , t h a t is, toi ( ' l ) ( ' 2 ) l £ ( » - ] ) ( M - 2 ) 1 d b l i t

q ( ) ( )£( ) (/- l ) ( r + r ' -2 ) + l, or £ (»- ] ) ( M - 2 ) + 1 double points.

Rational Transformation betiveen Two Spaces. 143

n

« i

a*

a*

°7

a9

2

3

1

3

41

1

4

601

1

330

2

8001

1

5

3310

2

0600

3

10000

I-H

1

6

r-t

4200

2

41300

3

34010

4

1200001

1

232100

2

7

034000

3

503100

4

—s

350010

5

n

a2

« 8

"4

"5

a6

<*7

14000001

1

3230100

2

1322000

3

8

0070000

4

3600010

5

6013000

6

0520

r-t

00

7

2051000

8

3303000

9*

160000001

1

41400100

2

23

r-t

21000

3

04040000

4

9

37000000V-^y-

5

»-\70031000

^»

6

\~

4300100

7

- • - • >

30411000

8

<~03311000v^-

9

1330000

10

n

« i

a,a8

"4

°5a0

n 7

«t t

1800000001

1

50500

0

r-t

00

2

pH

402o

0000

3

02231000

0

4

007001000

5

38000001u

f>

800130000

7

234000100

8"

10

403201000

0

13220

rH

000

10

213

i—t

20000

^>11

330301000

12

331030000

13

300G00000<—>14

060030000

15

015020000

IG

102500000

17

144 Professor Cayley on the

«=p

«1

«a

Op-l

« P

a2p-'

n = 2p

— ^= 2p-2= 0= 0

i = 1

2p-20

r-i

30

J

= 2p + l

n2p-l —

= 3= 2 p -= 0= 0

1

2p-i0310

n = Bp

«1a 2a3

ap

°P+Ia?P-ia3p-J

— 1= 4= 2p-3= 0= 0= .0= 1

2p-3004110

i

a, = "iV—\

2p-\001410

1

«i = 2

a2 = 3

a3 = 2p—2

Op = 0

<Vi = 0

«ip = 0

a3^-j= 1

w = 3p + l

2p-200

3

2

1

0*

ra\

« 3

ap

Q2p +

a3p-i

1

= 5= 2 p - l= 0= 0= 1

\2p-l

0. 5

10

i

ia, =

a3 =

«3 = 2

ap =

a P + i =a?p+i =a3p-l =

V.

3

2

J P - 100

0L

n = i

2p—l

0023

10

J

!p+2

a, =

aa =

a , = S

ap+i =

a2p =

a3p-l =

05

lp-200

1Y

2p-i0051

0

n =ta, =

a2 =

a, - =

°4 =

a, =

°P»I =

«2/- l =

a«P =

1

3

22JJ-3

000

01

2p-30

003

1

1

20

j

a, = 2n3 = 5a4 = 2 p - 4«„ = 0a p , i = 0

«3p-l= 0

«4i,-4= 1T

2p-40

05210

i

r«i =

a, =

a4 = 2

a ;>- l =

a p =

a 2 p =

3

3

•jp — 2

0001

2p~20

0

1

3

3

0J

ra, =

a3 =

a4 = 2

ft/,-1 =aP =

"3,, =

v«4,-4 =

G

1

p-200

0

1

2p-20

016

1

0


n =

a, = 0

a2 = 3a3 = 3a4 = 2 p - 3ap = 0 .aP+i = 0

a2p = 0a 4p-3= 1

000l330

do =

231-200001

2^-200031210

Cta =

2/?-3004310

al

a2

a3

«4

aP

Op,

°"ip

= 1q

= 2= 2p-2= 0

,= o= 0

a 2 p + l = 0

«4p-2= 1

y^

2p-200013120

33

o —

0001

al

a3

a4

aPa P + l

a3pa... o

= 4= 3= 2p-2= 0= 0= 0= 1

2p-2003410

n = 4/J + 3

VOL. in.—NO. 30.

146 • Prof. Cayley on the

42. The system (au a2... <*nL\) geometrically determines completelythe system (a,', a2'... a,',_i); it ought therefore to determine it arith-metically; that is, given the one series of numbers, we ought to be ableto determine, or at least to select from the table, the other series ofnumbers. Cremona has shown that the two series consist of the samenumbers in the same or a different order. By examination of the tables,it appears that there are certain columns which are single (that is, noother column contains in a different order the same numbers), othersthat occur in pairs, the two columns of a pair containing the samenumbers in a different order. Where the column is single, it is clearthat this must give as well the values of (a'b a2... a',,^) as of(a,, a2... a,.j). Where there is a pair of columns, as far as Cremonahas examined, if the one column is taken to be (a,, a2 . . . an_j) theother column is (a',, a'i...a'n_1) ; it appears, however, not to be shownthat this is universally the case; viz., it is not shown but that thetwo columns, instead of being reckoned as a pair, might be reckonedas two separate columns, each by itself representing the values aswell of (ob aa.. . a,,.,) as of (aj, aa... a^.j) ; neither is it shown thatthere are not, in any case, more than two columns having the samenumbers in different orders. It seems, however, natural to supposethat the law, as exhibited in the tables, holds good generally; viz.,that the tables contain only single columns, each giving the values aswell of (crb ai...an_,) as of («i, aa...a|,_,) ; or else pairs of columns,one giving the values of (au a2...an.x), and the other those of(oj, a'j...aj,_i) ; or, say, that the partitions are either sibi-reciprocal, orelso conjugate.

43. Assuming that the two systems (ab aa...an_l) and (a',, aj...aj,_,)are each known, there is still a question of grouping to be settled;viz., the Jacobian of the first plane consists of ai lines, aa conies,... a'H_x

unicursal (n—l)-thics; each line, each conic, &c, passes a certainnumber of times through certain of the points au aa.. .cr,,_i: but throughwhich of them ? For instance, each of the ai lines will pass throughtwo of the points au a.i...an_1: will these bo points a,, or points a2,<fec, or a point a, and a point au &c. ? The mere symmetry of thedifferent groups of points determines certain conditions of the solution;*for instance, if any particular one of the ai lines passes through twopoints ar, then each of the ai lines must pass through two points ar;and since the points u, mo symmetrical, we must in this way use allthe pairs of points ar; that is, if ai = \ar (ar-f 1), but not otherwise,

* I t iH by such considerations of symmetry that Cremona has demonstratedthe beforo mentioned theorem of tho identity of the nunibors (a1( a3...an-1) und(a,', a/...a',,_i)

Rational Transformation betiveen Two Spaces. 147

it may he that each of the a\ lines passes through two of the points ar.In the case of an equality ar = a, we could not hereby decide whetherthe line passed through two points ar or through two points a,. So,again, if any one of the a\ lines pass through a point ar and a point a,,then each of the a\ lines must do so likewise, and we must hereby ex-haust the combinations of a point ar with a point a,; viz., the assumedrelation can only hold good if a\ = ara,. Similarly, each of the ojconies will pass through five of the points au aa...an_i; each of the a'2nodal cubics will pass twice through one (have a double point there)and through six others of the points au a3...an.i: which are the pointsso passed through ? I do not know how a general solution is to beobtained, but most of the cases within the limits of the foregoing table •have been investigated by Cremona. The results may conveniently bestated in a tabular form; the tables exhibit in the outside upper linethe values of <xj, ct3...an_i, and in the outside left-hand line the valuesof o'i, aj...a'n_i, and they are to be read as follows: Each of the

5a\ lines ")a2 conies?- passes ( ) times through ( ) of the points o^ ttt...an-i

Ac, Jrespectively; the numbers in the table being those of the points passedthrough, and the indices in the table (index = 1 when no index is ex-pressed^ lowing the number -of lines of passage, that is, showingwhether the point is a simple, double, triple, &c, point on the curvereferred to.

44. Thus the last of the tables n = 6 gives the constitution of theJacobian of the first plane, where the principal system is (3,4, 0,1,0);and it is to be read:—Each of the 4 lines passes through 1 of the points ax and through

the point a4;The 1 conic „ „ 4 of the points a2 and through

the point a4;Each of the 3 cubics „ „ 2 of the points au 4 of the points

n2, and twice through the pointai (that is, at is a double pointon each cubic).

It is hardly necessary to remark that the tables are sibi-reciprocal, orelse conjugate, as appears by the outer lines of each table.

I, 2


TABLE n — 2.

a\ = 4

"aII1

1

4

1

1TABLE n — 3.

" 33 = l

«1 «2 a3

II II II6 0 1

n, a2

II II3 3

1

G

1

I2

3

3

0

2

2

3 TABLES w = 4.

a3 = v

o;=i

'1IIII8

a 2IIII0

"3

0IIII1

TABLES

II IIII II3 3

» =

IIII1

:5.

a*IIII0

IIII0

IIII6

"3IIII0

«4IIII0

1

8

1

Is

3

3

1

0

1

3

1

3

3

1

1

1"

0

6

0

0

5

Rational Transformation between Two Spaces.

TABLES n = 6.

Q\ Qj 03 o^ or 5

149

10 0 0 0II II4 2 0 0

a,' =10

c4= 0

a 3 = 0

• A

a ' 5 = 1

1

10

1

I4

1

4

2

0

0

1

3

4

2

2

•

a i ai Oj a4 asII II II II II4 1 3 0 0

a. 1 cij tig a 4

II II II II3 4 0 1

a j= 3

ai = 4

a3= 0

a i= 0

1

4

1

1 .

2

3

32

•

4

1

3

0

0

1

2

-4

4

1

1

I2

• Road, " Each of tho two cubics passes tlirough tho point ab tho four points aj,and, (I2,1), twice through ono and onco through the other of tho points oj."

45. It is to be remarked upon the tables—first, as regards the lines:if wo add the numbers in each line, reckoning mp as mp, (that is, eachmultiple point, according to the number of branches through it,) thesums for the successive lines are 2, 5, 8, 11, 14, &c; that is, each linepasses through 2 points, each conic through 5 points, each cubicthrough 8 points, each quartic through 11 points, &c. But if we addthe numbers reckoning m" as m.^p(p + l), (that is, each multiplepoint according to its effect in the determination of the curve,) thenthe sums are 2, 5, 9, 14, 20, &c, that is, all the curves are completelydetermined, viz., the line by 2 conditions, the conic by 5 conditions,the cubic by 9 conditions, &c. Secondly, as regards the columns, iffor any column, reckoning mp as mp, we multiply each number by thocorresponding outside left-hand number, add, and divide tho sum by thooutside number at the head of the column, tho successive results aro


2, 5, 8, 11, 14, <fed.; this merely expresses the known circumstancethat the Jacobian passes 3r—1 times through each point ar.

46. The analogous tables showing the passage of the Jacobianthrough the principal system, in the solutions belonging to certainspecial forms of n, are

TABLE n=p.

2/)-2

= 2p—2

= 0

= 1

= 3

0

a\ = 2 ^ -

aj = 0

«; = 3

a ; , , = 1

ai,»-i= 0

1

2p-2

1

1P-»

TABLES n = 2jp.

Ul a2 ap-\ «p a2p-2 «1 a2 °p-l ap a2p-J

II II II II II II II II II II3 2jo-2 0 0 1 2p-2 0 1 3 0

o ; = 3

a2 = 2p—2

ap-i = 0

np = 0

a 2 p - 2 = 1

1

2

2p-2

2p-2

1

1P-2

1P-1

1

2p-2

1

2

3

3P-1

TABLES »t=

II II II II II3 2 ; > - l 0 0 1

1

3

1

2 , - 1

2p-\

1

IP-.

1"

«; =s 3

0 3 = 2p—1

= 0

aptl= 0

a2p-lII II II II II

2j) - l 0 3 1 0

1

1

2 i » - l

3

3P-1

1

1

1".


TABLES n = Sp.

«2p-l «3p-S

III II II II II4 2^-3 0 0 0

a; = 2p-3

a'z = 0

»; = 4

a 3n-3= 0

1

1

3

4

4

1

2^-3

2p-3

(2^-3)2

1

p - i

1P

J2p-3

Ol a2 a3 ap Qp+1 a2p-l a3p-3II II II II II II II

2p-3 0 0 4 1 1 0

«; = I

O'2 = 4

a[, = 22?—3

a, = 0

o ; . , = o

-i= 0

1

2p-3

3

4

1

1

1

1"

1

1

l a

\lP-3

152

a\ = 2p—2

= 0

= 0

a' ! s= 1

a'p = 4

; p . 3 = o

a', = 4

aj = 1

ai = 2p—2

«;.! = o

•; = o

2P = 0

Prof. Cayley on the

TABLES n = 3p.

ct8 a 8

II II1 2p-2

ap_,II0

II II0 0

a3p-8II1

1

4

1

1

1

2,-2

2p-2

1

! M

I -

«1 Oj «3 a p - l ap a2p aSp-8II II II II II II II

2p-2 0 0 1 4 1 0

1

2^-2

1

1

4

4

* - -

1

1

la


47. The before mentioned theorem, that (au a,... a,,.!) and (a'u a\... a|_i)are the same series of numbers, of course implies 2ar =; Sa'r; this re-lation Cremona demonstrates independently, by consideration of thepencil of curves (aX + bY + cZ)+d(alX+blY+c1Zi) = 0, (0 a variableparameter,) which corresponds in the first plane to the pencil of lines(ax'+by'+cz') + d (fleas'+&iy'+Ciz') = 0, which pass through a fixedpoint (ax'+by' + cz' = 0, a^x'+biy+ctz' = 0) in the second figure. Ingeneral, in the pencil U+ffV = 0 (U, V given functions of the order n)there are 3 (n—I)2 values of 6, each giving a nodal curve. But in thepresent case each of the curves U = 0, V = 0 has multiple points at theprincipal points ar of the first plane: the question is to obtain the num-ber of values •which give a curve having one new double point; andthis.is found to be = 3 (n—1)2-2 ( r - 1 ) (3r + l ) ar. We have2r2ar = n*—1, 2rar = 3n—3 ; or, substituting, the value of 8 is = 2ar.But the curves which have an additional double point are those whichcorrespond to the lines which in the second figure pass through one ofthe principal points ar; viz^ these, are the lines drawn from the point(ax'+by'+cts'=. 0, axx'+bx\j + c&' = 0) to the several principal points a'r;and the number of them is = See',. We have thus the required relation

The Quadric Transformation between Two Planes.

48. This is of course given by what precedes. The principal systemin each plane is a set of three points; and the Jacobian of the sameplane is the set of three lines joining each pair of points; that is, thethree Hues of either plane are the principal counter-system of the otherplane. But to give the analytical investigation directly: taking thecoordinates (as, y, z) to refer to the principal system of the first plane(viz., taking the three points to be the vertices of the triangle formedby the lines x=0, ?/=0, z=0), then X=0, Y=0, Z = 0 being coniesthrough the three points, the functions X, Y, Z will be each of them ofthe form fyz+gzx+hxy; x', y', z' being proportional to three such func-tions, there will be linear functions of x', y\ z proportional to yz, zy, xy ;or taking these linear functions of the original (»', y\ z') for the coor-dinates (a?', y\ z') of a point in the second plane, the formulro of trans-formation will be x : y : z •=• yz \ zx : xy, and we have then converselyx : y : z = y'z : z'x': xy'; that is, the formulce for the transformation inquestion are

x : y : z = yz : zx : xy,. and x : y : z = y'z': z'x': x'y'.We at once verify a posteriori that the Jacobian in the first plane isxyz = 0, and that in the second plane is x'y'z — 0.

The equations may be written, , - , 1 1 1 T I l l

x : y : z = - : - : - and ;c : y : z : = -7 : — : -7,x y z x y z

154 Prof. Cay ley on tJw

or, if we please, xx—yy'—zz: the transformation's thus given as aninverse transformation.

[49. With respect to the metrical interpretation and actual constructionof the transformation, it is to be observed that if as, y, z be taken to beproportional (not to given multiples of the perpendicular distances, but)to the perpendicular distances of P from the sides of the triangle in thefirst plane, and similarly x\ y', z to be proportional to the perpendiculardistances of V from the sides of the triangle in the second plane, thenin general the equations of transformation must be written, not as above,

but in the form — = ***- = —, involving arbitrary multipliers f:g:h.f 9 "•

We may imagine in the second plane a point P" determined by coor-dinates (*",y", z"),—the same coordinates as (as', y\ z'), that is, propor-tional to the perpendicular distances of P" from the sides of the trianglein the second plane,—which point P" corresponds homographically to P

in such wise that '— : ^ : — = as": y": z". We have then, in the secondf 9 h

plane, the two points P', F ' corresponding to each other in such wisethat x'x"= y'y"— zz"\ and either of these points being given, the othercan at once be constructed ; viz., it is obvious that, joining P', P" withany vertex, say A', of the triangle A'B'C, the lines A'P', A'P" are equallyinclined to the bisectors of the angle A'; and consequently, P* beinggiven, we have the three lines A'P", BTP", C'P" intersecting in .a com-mon point P", which is therefore determined by means of any two ofthese lines. We have thus a geometrical construction of the transfor-mation between P and P'.]

50. The analysis assumes that the principal points A, B, C of thefirst figure are three distinct points; but they may two of them, or allthree, coincide. In the first case, say if B, C coincide, the line BC isstill to be regarded as having a definite direction; and taking x—0 forthis line, y = 0 for the line joining A with (BC), and 0=0 an arbitraryline through A, the functions X, Y, Z will be each of them of the formhif + 2gzx+21uty; and replacing, as before, the original x\ y\ z' by linearfunctions of these quantities, these linear functions being taken for thecoordinates (as', y\ z'), we may write x': y \ »'= •//*: xy : xz. Formingthe converse system, the equations for the transformation are

x : y : z' == y1 : xy : xz, and as : y : z — if1: xy : x'z\so that the points A', B', C in the second plane are related as the pointsin the first plane; viz., B', C coincide, the line B'C being definite.

It is easy to verify that the Jacobian in the first plane is a:?/2 = 0, andthe Jacobian in the second plane is x'y* = 0.

51. Secondly, if A, B, C all coincide, these being however consecutive


points on a curve of finite cnrvatnre, or say on a conic; then, taking05=0 for the tangent at (ABC), a = 0 for any other tangent, and y = 0for the chord of contact, the functions X, Y, Z will be of the formax2+b(yi—zx) + 2hxy ; whence we may write as' :yf :z' = a? :xy : y*—xz.Forming the converse equations, the equations of transformation are

a': y: z' = #a : xy : y*—xz, and x : y : z = as'2: x'y': y'a—asV;so that the points A', B', C in the second plane are related as those ofthe first plane; viz., they are the consecutive points of a curve of con-tinuous curvature.

We may verify that the Jacobian of the first plane is a?8 = 0, and theJacobian of the second plane as'8 = 0.

The Lineo-linear Transformation between two Planes.

52. We have two equations of the form(a, ...Jas, y, 2$>', */', z ) = 0,(a,,... Jas, y, z~$x\ y\ 0 = 0;

writing these in the form

where (P, Q, R-, Pi, Qi, Ri) are linear functions of (aj, y, z), we have

x\y\z' proportional to II £ % ^ II ," rutyu Kill

that is to X : T : Z, where X=0 , Y=0 , Z = 0 are conies each passingthrough the same three points in the first plane.

And conversely, writing the equations in the form

where (P*, Q', R', Pi, Qi, Ri) are linear functions of (x\ y, z), wehave

that is to X', Y', Z', where X'=0, Y'=0, Z'=0 are conies each passingthrough the same three points in the second plane.

53. The lineo-linear transformation is thus the same thing as thequadric transformation. It is, moreover, clear that the equations must,by linear transformations on the two sets of variables respectively, andby linear combination of the two equations, be reducible into formsgiving the before-mentioned values of x : y : z and x : y : z respec-tively. Thus, in the general case, where in each plane the three pointsare distinct points, the lineo-linear equations will be reducible to

n'x—yy' — 0, xx—zz = 0 ;


in the case where B, C in the first plane, and B', C in the secondplane respectively coincide, the forms will be

vx-yy = 0, yz'—y'z = 0;and in the case where A,B, 0 in the first plane, and A', B', C in thesecond plane respectively coincide, the forms will be

xy—yx=0, %z'—yy'-\-zx' = 0.The determination of the actual formulas for these reductions would,

it is probable, give rise to investigations of considerable interest.

The General Rational Transformation between Two Planes (resumed).

54. Consider, as above, the first plane or figure with a principalsystem (cij, aa... an_i), and the second plane or figure with a principalsystem (aj, a2... a,',_i). To a line in the second plane there corres-ponds in the first plane a curve of the order n passing 1 time througheach of the points au 2 times through each of the points ai} 3 timesthrough each of the points a8, &c.; or, as we may write this,

First figure. Second figure.Points aj aa a3 ... an.l Points a\ a2- a3 ... a'n.l

T 2 3 n~^T\ 0 0 0 n~^l-]•> 9 o „ -i c u r v e n (\ n „ i c u r v e

1 2 3 « - l l o r d e r 0 0 0 » - l l o r d e r

1 i 3 n - l | n • : • 0 i l

o r d e

l | n • : • 0 i

viz., the l's denote the number of times which the curve of the order npasses through the several points ax respectively; the 2's the numberof times which the curve passes through the several points a8 respec-tively ; and so on.

55. We may, in the second figure, in the place of a line consider acurve of the order 1c. If the equation hereof is ( # ]JV, y\ z')k' = 0,then the corresponding curve in the first figure is ( # ]JX, Y, Z)*" = 0 ;viz., this is a curve of the order k=nh'. If, however, the curve in thesecond figure passes once or more times through all or any of thepoints a'i, n't... a'n-it then there will be a depression in the order of thecorresponding curve in the first figure ; and, moreover, this curve willpass a certain number of times through all or some of the points«i, a2> a3 ••• a.»-i« The diagram of the correspondence will be

First figure. Second figure.au a2, as ... a,,.! aj, a'2, a3 ... a|,_!

ax ti% 03 d '


where a^ bu Ci... denote the number of times that the curve of theorder k passes through the several points nj respectively, (viz., thenumber of the letters au bu cx... is = au any or all of them being zeros,)Oj, 62, c8 ... the number of times that the curve passes through theseveral points a2 respectively, (viz., the number of the letters c , b2i c2 ...is = a2, any or all of them being zeros,) and so on; and the like for thecurve in the second figure.

56. By what precedes, it is easy to see that, if the curve Jc passesthrough a point a',, then the curve k throws off a line, and the de-pression of order is = 1; so, if the curve passes 2 times, 3 times,...or at times through the point in question, then the curve throws off tholine repeated 2 times, 3 times, ... a\ times, or the depression of orderis = 2, 3 ... or a'ii and the like for each of the points a\; so that,writing for shortness a\ + b'i-\-c'l+ ... = SaJ, the depression of order onaccount of the passages through the several points oj is = 2aj. Simi-larly, for each time of passage through a point «r2, there is thrown off aconic; or if a'2 + b2 + ... = 2a2, then the depression of order is = 22c^,and so on ; and the like for the figure in the other plane; and we thusarrive at the equations

h = k'n—2Jc — hi—2

57. The simplest case is when the curve k' does not pass throughany of the points a'lt aa, ... a'n_i. We have then

a = 6i = ci... = 0, oi = &a... = 0, e4.i = &;,_,... = Ofconsequently k = k'n. And, moreover, it is easy to see that

so that the correspondence isFirst Figure. Second Figure.

a l » a 2 l O8> • • • a » - l a b tt2» °S> • • • O H

Jc' 2k' 2Jc (n-l)fcS curve 0 0 0 (T) curveJc Zk' BJc (n-l)Jc[ order 0 0 0 0 f order: : : : ) k = nk' : : ; : ) k'.

We haveSai = k'alt Saj = 2&/a18, 2ai (ai— 1) = k'(k'— 1) a!, &c,

and the formulae for k, k' becomek = k'n, k'= kn—k'l^ + Aaz... +(n—l)2an.1( ;

viz., the second equation is here k'— hi— A;'(?ia—1) ; that is, k>n2 = kn,agreeing, as it should do, with the first equation.

58. Moreover, the deficiency-relation is

l) + 2k'(2k'-l)...+^lk'frolic-I)\= Kt f - ! ) (* ' -2 ) ;

158 . Prof. Cayley on the

or, what is the same thing, this is(nk'-l) («&'-2)-(/fc'-l) (&'-2) =

The right-hand side is

and we have thus the identical equationink*-1) (nk'~ 2) - (kf-1) (Jc'~ 2) = (n-\) k' j(»+1) A;'- 3 (.

59. It should be possible, when the nature of the correspondencebetween the two planes is completely given, to express each of thenumbers alt bu Ci, ... a,,_i, bn.u ... in terms of &', a',, 6 , c\t. ... a,',_i,&n_ii . . .; and reciprocally each of the numbers a'lf b\, cu ... ai,_i, 6j,_!, ...in terms of &, a-i, 6,, Cj, ... an_i, &„_;, ... ; thus completing a system ofrelations between the two sets

(&, au bu ... aH.h &„_!, . . .) , (h\ a'u b\, ... a|,_!, b'n.it . . . ) ;

but even if the theory was known, there would be considerable difficultyin forming a proper algorithm for the expression of these relations.

60. The two curves must have each of them the same deficiency. Itis to be noticed, that if the curve in the first plane passes any numberof times through a point P, which is not one of the points au a2, a3, ...or an_b then the corresponding curve in the second plane will pass thesame number of times through the corresponding point P', which pointwill not be one of the points a\, a'2t ... a'n_i. The points P, P ' willtherefore contribute equal values to the deficiencies of the two curvesrespectively; so that, in equating the two deficiencies, we may dis-regard P, P', and attend only to the points au a8, ... a,,_, of the firstplane, and aj, a2, ... a'n_x of the second plane. And the requiredrelation thus is

£ ( £ _ ! ) ( £ - 2 ) - 2 * {Oi (Oi

= * (* ' - ! ) (& ' - 2) - 2 f {a; ( a ; -

61. In the case of the quadric transformation w=2, we have in thefirst plane the three points «i, say these are A, B, C; and in the secondplane the three points a'u say these are A', B', 0'. And if in the firstplane the curve of the order k passes a, 6, c times through the threepoints respectively, and in the second plane the corresponding curve ofthe order k' passes a', fe', c times through the three points respectively,then it is easy to obtain

k'= 2k—a—b—c,a'= k—b — c,

c'= k — a—b.

k = 2k'- a'~ &'-a = k'— V— c\b = k'— c — a,c = k'— a — V.

Rational Transformation between Two S}wccs. 159

The Quadric Transformation any number of times repeated.

62. We may successively repeat the quadric transformation accordingto the type—

First Fig. Second Fig. Third Fig. Fourth Fig.A, B, C A', B', C

D', E', F D", E", F"G", H", I" G'", H"', I'"

viz., in the transformation between the first and second figures, theprincipal systems are ABC and A'B'C respectively; in that betweenthe second and third figures, they are D'E'F and D"E"F" respectively;in that between the third and fourth figures, they are G"H"I" andG'"H"T"; and so on. And it is then easy to see that between the firstand any subsequent figure we have a rational transformation of theorder 2 for the second figure, 4 for the third figure, 8 for the fourthfigure, and so on.

63. But to further explain the relation, we may complete the dia-gram, by taking, in the transformation between the second and thirdfigures, A", B", C" to correspond to A', B', C; similarly, in that betweenthe third and fourth, A'", B'", C" to correspond to A", B", C"; andD'", E'", F'" to correspond to D", E", F". And so in the transforma-tion between the second and third figure, we may make G', H', I' cor-respond to G", H", I", and between the first and second figures makeD, E, F correspond to D', E', F , and G, H, I to G', H', I ' , the diagrambeing thus:

First Fig. Second Fig. Third Fig. Fourth Fig.

A,B,C A', IT, C'l A",B",C" A'", B"\ C"

D, E, F |i^_E\JI JD", E", F' | D'", E'", F"G, H, I G', H', I' |G"~~H",r' G"',H'",r"l

Observe that in the principal systems (for instance, A, B, C andA', B', C) the points A, B, C correspond, not to the points A', B', C,but to the lines B'C, C'A', A'B' respectively; and so in the other case.

64. Consider now a line in the first figure: there corresponds heretoin the second figure a conic through the points A', B', C; and to thisconic there corresponds in the third .figure a quartic curve passingthrough each of the points A", B", C" once, and through each of thepoints D", E", i1" twice. And conversely, to a line in tho third figurecorresponds in the second figure a conic through the points D', E', F';nnd hereto in the first figure a quartic through tho points D, E, F, onco


and through the points A, Bj C twice ; that is, we have between thefirst and third figures a quartic transformation wherein o, = aa = 3 anda'i = a i = 3 , or say a quartic transformation 3i32 and 3I3a. In likemanner, passing to the fourth figure, to a line in the first figure corre-sponds in the fourth figure an octic curve passing through A'", B'", C"once, through D'", E'", F " twice, and through G'", H'", I'" four times;and conversely, to a line in the fourth figure there corresponds in thefirst figure an octic curve passing through the points G, H, I once, thepoints D, E, F twice, and the points A, B, C four times; that is, be-tween the first and fourth figures we have an octic transformation,wherein a1 = aa = a4 = 3, a\ = a3 = a = 3, or say a transformation,order 8, of the form 31323< and 3i3234. And so between tho first andfifth figures there is a transformation, order 16, of the form 313J3438

and 3i323i38.

65. It is, moreover, easy to find the Jacobians or counter-systems inthe several transformations respectively. Thus, in the transformationbetween the first and second figures, in the second figure the Jacobianconsists of 3 lines such as B'C (viz., these are, of course, the lines B'C,C'A', A'B'). Hence, in tho transformation between the first and thirdfignres, the Jacobian in the third figure consists of

3 conies B"C"(D"E"F"),3 lines D"E";

viz., one of the conies is that thi'ough the five points B", C", D", E", F",one of the lines that pass through the two points D", E". And so inthe fourth figure, the Jacobian consists of

3 quartics B"/C"/(D"'E"/F"), (G '"HT) 2 ,3 conies D'"E'" {G""E"T')U

3 lines G'^H'";viz., one of the quartics passes through B'", C"; through D'", E*", F"'each once, and through G'", H"', I"' each twice. And so in the fifthfigure the Jacobian consists of

3 octics B""C"//(D""E""F'"). (G^'H""!""), (J""K""L""),,3 quartics D""E"" (G""H"T'")i (J""K""L"")2,3 conies G""H"" (J""K""L""),,3 lines J""K"",

and so on.

GG. Tho conditions are in each case sufficient for tho determinationof tho curve. This depends on ilio numerical relation

4 + 3 ( 1 . 2 + 2 . 3 + 4 . 5 + 8 .9 . . . +2*(2' + l ) j = 2*+1 (2' l I + 3).The term in {{ is ' 1+4 + 16 ...+22*

+ 1 + 2+ 4...+2%


2 2 s + 2 - l , 2 8 t l - l

which is • = i [2M*8-1 + 3 ( 2 8 t I - l ) ]

= £[2M+J + 3 . 2 ' + 1 - 4 ] ;

and the relation is thus identically true.

67- Conversely, in the transformation between the first figure andthe several other figures respectively, the Jacobian of the first figure is

o r ATJ ") for order 2, between firsto lines Ar> > '

) and second figures;and so 3 conies DE (ABC)i ") for order 4, between first

3 lines AB 5 and third figures ;3quartics GH(DEF)1(ABC)2 ^„ • TVCI / A T»n\ I f°r order 8, between first3 conies DE (ABC)! ^ '£ .. A-r> and fourth figures:3 lines AB J b '3 octics JK (GHI), (DEF)2 (ABC)4 -j3 quartics GH (DEF), (ABC)-: I for order 16, between first3 conies DE (ABC), | and fifth figures;3 lines AB Jand so on.Special Cases—Reduction of the General Rational Transformation

to a Scries of Quadric Transformations.68. It was remarked by Mr. Clifford that any Cremona-transformation

whatever may bo obtained by this method of repeated quadric trans-formations, if only the principal systems, instead of being completelyarbitrary, are properly related to each other. To tako tho simplest in-stance ; suppose that we have

First figure. Second figure. Third figure.A,B,C A',B',C B",C"

IIE, F D', E', F' D", E", F"

viz., in the transformation between the first and second figures, wo havethe principal systems ABC and A'B'C (arbitrary as before); but in thetransformation between tho second and third figures, tho principalsystems are D'E'F' and D"E"F", where D', instead of being arbitrary,coincides with A'. And wo then have B", C" in the third iigure cor-responding to B', C in the second figure, and E, F in the first ligiuocorresponding to E', F' in the second Iigure. This being so, to a lino inthe first figure corresponds in the second figure a conic through A', B', C.But A'=D'; viz., this conic passes through a point D' of tho principal

VOL. in.—NO. 31. M


system of the second figure, in regard to the transformation betweenthe second and third figures. That is, (k, a, 6, c referring to the secondfigure, and k',a,b\c to the third figure, k — 2, a = 1, b = 0, c = 0, andtherefore &'= 3, a '= 2, &'= 1, c'= 1,) corresponding to the conic we havein the third figure a curve, order 3 (cubic curve), passing twice throughD", but once through E" and F" respectively; this cubic curve passesalso through the points B", C" which correspond to B', C respectively;that is, cubic passes through E", F ' , B", C" each 1 time

„ „ D" 2 times;or, corresponding to a line in the first figure, we have in the third figurea curve, order 3, passing through four fixed points each 1 time, andthrough one fixed point 2 times. That is, we have n = 3, a', = 4, a'a =z 1.And in the same manner, to a line in the third figure there correspondsin the first figure a cubic through four fixed points (viz., B, C, E, F)each 1 time, and through one fixed point, A, 2 times; so that alsoaj=4, a 2 = l . The transformation is thus of the order 3, and the form4, la and 4t la (this is in fact the only cubic transformation; see theTables, ante, No. 41).

69. Mr. Clifford has also devised a very convenient algorithm forthis decomposition of a transformation of any order into quadric trans-formations. The quadric transformation is denoted by [3], the cubiotransformation by [41], the quartic transformations by [601], [330],the quintic ones by [8001], [3310], [0600], and so on; see the Tablesjust referred to. (This is substantially the sam^ as a notation em-ployed above, the zeros enabling the omission of the suffixes; viz.,[8001] = 8i 14; and so in other cases.)

70. The foregoing result is represented thus, [4,1] = [3JO, 0,1),which I proceed to explain. Consider in the first figure a line; thesymbol [3] denotes that in the second figure we have a conic withthree points (a',). We are about to apply to this a quadric transfor-mation ; (0,0, 0) would denote that the three points of the principalsystem in the second figure were all of them arbitrary; (0, 0,1) thatone of these points was a point aj ; (0,1,1) that two of them werepoints o',; (1,1,1) that all three of them were points aj; (0,0,2) woulddenote that one of the points was a point a'a; only in the presentcase we can have no such symbol, by reason that there are no points a\.Hence [3J001) denotes that the conic has applied to it a quadrictransformation such that, in the transformation thereof, one pointof the principal system coincides with one of the points (a',) onthe conic. To [3], qua quudriu transformation, belongs the number2; and from 2, (001) we derive 3, (112), [in general Jc, (a, 6, c)gives k\(a,h',c), where k'=2k—a—b—c, a'=fc—b—c, &'=&—c—a,c—k—a—6]. k = 2 corresponds to a symbol [3] of one number,k'— 3 to a symbol of two numbers; viz., we change [3] into [30];


we then, in the symbols (112) and (001), consider the frequenciesof the several numbers 1, 2 ... taking those in the first symbol aspositive, and those in the second symbol as negative; or, whatis the same thing, representing the frequency as an index, we haveI2 21, I"1; or, combining, I*"12l; these indices are then added on to thenumbers of [30] ; viz., the index of 1 to the first number, the index of2 to the second number (and, in the case of more numbers, so on):[30] is thus converted into [41], and we have the required equation

[41] = [81001),where the rationale of this algorithmic process appears by the expla-nation, ante, No. 68.

71. As another example take[8001] = [601^003).

To [601], qua quartic transformation, belongs the number 4 ; andfrom 4,(003) we form 5,(114); where the 5 indicates that [601] is to .be changed into [6010] ; then (114), (003), writing them in the formV2°S-l4>\ show that to the numbers of [6010] we are to add 2,0, —1,1;thus changing the symbol into [8001], so that we have the requiredrelation.

72. Mr. CliflFord calculated in this way the following table, showinghow any transformation of an order not exceeding 8 can be expressedby means of a series of quadric transformations; the symbols Cr. tt,Cr. 4 . 1 ; 4 . 2 , &c, refer to the order and number of Cremona's tables,ante, No. 41.

Cr. 3 . [41] = [ 3 p 0 1 )

Cr. 4 . 1 = [601] = [41J002) = [ 3 p 0 1 £002)

4 .2 = [330] = [3^000)

= [41 $011) = [3$001][011)

Cr. 5 .1 = [8001] = [601^003) = [3p01£002£003)

5.2 = [3310] = [41 $001) = [3 ([001 £001)

5.3 = [0600] = [330$lll) = [3$000£lll) = [3$001 £011£lll)

Cr. 6.1 = [10,0,0,0,1] = [8001$004) = [3$00l£002£003£004)6.2 = [14200] = [330][011) = [3$000£011) = [3$00l£011£011)6 .3= 41300] = [6011011) = [3$001£002£011), =

= [3310$022) = [3$001£001£022)J0.4 = [34010] = [330$002) = [3 $000 £002)

= [3310$013) =M2

IGt Prof. Cayley on the

Cr. 7.1 = [12,00001] = [10,0001 $005) = [3£001][002$003$00]£005)7.2 = [330p01) = [3p00£001) = [232100]7.3 = [034000] = [3310 <[111) = [SJOOIJOOIJIII)7 .4= [503100] = [6013:001) = [3$0015; 002^001)7.5 = [350010] = [3310p03) = [3J001][001][003)

Cr. 8.1 = [14, 000001] = [3$001 500250035004p05][006)8.2 = [3230100] = [3310p02) = [3J001 $001 $002)8.3 = [1322000] = [SSIOJOII) = [33:0015001][011)8.4 = [0070000] = [0340003:222) = [33:001£001$lll$222)8.5 = [3600010] = [340103:004) = [3303:002][004)

8.6 = [6013000] = [6013:000) = [33:00150025000)8.7 = [0520100] = [06003:002) = [3][000][lll][222)8J3 = [2051000] = [413003:112) = [33:00150005112)8.9 = [3303000] = [3 <[ 000 5 000 5 000)

73. The reduction as above of a transformation to a series of quadrictransformations, enables the determination of the reciprocal trans-formation; or, what is the same thing, the determination of theJacobiiin of the first figure ; see the example, mite, No. Q7, "where itappears that tho reciprocal transformation of [41] is [41]. But I donot sec any easy algorithmic process for the determination of thereciprocal transformation, or still less any general form in which theresult can be expressed ; and I do not at present pursue the inquhy.

THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES.

74. The general principles have been already explained: tho twosystems x : y : z : w'=X : Y : Z : W and x : y : z : w=X ' : T: 71: Wmust be derivable the one from the other; and starling with the firstsystem, this will be the caso if only the surfaces X = 0 , Y=0, Z=0 ,AV=O have a common intersection equivalent to ns— 1 points of inter-section, but not equivalent to a complete common intersection of ns

points. The last mentioned circumstance would arise, if the conditionof the common intersection should impose upon the surface more than£ («-f l)(»4-2)(u + 3)— 4 conditions; viz., the surfaces would then beconnected by an identical equation or syzygy aX+/3Y + yZ + 5W = 0.The common intersection is a figure composed of points and curves :say it is the principal system in tho first space ; tho problem is, toildermine a principal system equivalent to M3—1 points of intersectionbut such Uiiil the number of conditions to bo satisfied by a surfacepassing through it is not more than £ (n + l)(?i + 2)(?i+3) —4.

Batlonal Transformation between Two Spaces. 165

75. The following locutions are convenient. "We may say that thenumber of conditions imposed upon a surface of the order n whichpasses through the common intersection is the Postulatlon of thisintersection ; and that the number of points represented by the commonintersection (in regard to the points of intersection of any three surfaceseach of the order n which pass through it) is the Equivalence of thisintersection. The conditions above referred to are thus

Equivalence = n9—1,Postulation > £ (n +1) (n + 2) (n + 3) — 4.

76. It would appear by the analogy of the rational transformationbetween two planes, that the only cases to be considered are thoso forwhich Postulation = £ (n +1) (n+2) (n+3) - 4;but I cannot say whether this is so.

77. In the transformation between two planes, the two conditionslead, as was seen, to the result that the curve aX-H&Y-j-cZ = 0 isunicursal. I do not see that in the present case of two spaces, thetwo conditions lead to the corresponding result that the surfaceaX.+bY+cZ+dW = 0 is unicursal; that this is so, appears, however,at once from the general notion of the rational transformation. Infact, the equation in question aX.+bY+cZ + dW = 0 is satisfied byx : y : z : w = X': Y': 71: W Snd ax' + by' + cz' + dw = 0 ; the lastequation determines the ratios x': y : z : w in terms of two arbitraryparameters (say these are x': y' and x': z')} and we have then x : y : z : wproportional to rational functions of these two parameters ; that is, thesurface aX + bY + cZ + dW = 0 is unicursal. And similarly the surfacoaX' + bY' + cZ'+dW = 0 is unicursal.

78. In the most general point of view, the principal system willcontain a given number of points which are simple points, a givennumber which are quadriconical points, a given number which arecubiconical points, &c. &c., on the surfaces; and similarly a givounumber of curves which are simple curves, a given number whichare double curves, &c. &c, on the surfaces. But, to simplify, Iwill consider that it includes only points which are simple points,and a curve which is a simple curve on the surfaces : this curvomay, however, break up into separate curves, and wo thus, infact, admit the case where there aro any number of separate curveseach of them a simple curve on the surfaces. It is right to remarkthat we cannot assert a priori—and it is not in fact the case—that thoprincipal system in the second space will be subject to the like restric-tions : starting with such a principal system in tho first space, womay be led in the second space to a principal system including a curvewhich is a double curve on the surfaces ; an instance of this will infact occur.


79. It is shown (Salmon's Solid Geometry, 2nd ed., p. 283), that inthe intersection of three surfaces of the orders /*, v, p respectively, a curveof intersection of the order m and class r counts as m (/i + i'+p—2)—rpoints of intersection. For a curve without actual double points orstationary points, we have r = m (m—l)—2h, where h is the numberof apparent double points; or, substituting, we have the curve countingfor ra(/i + v+p—2)— m(m—l) + 2h points of intersection; this is infact a more general form of the formula, inasmuch as it extends to thecase of a curve with actual double points and stationary points. Or,what is the same thing, the three surfaces intersecting in the curve ofthe order m with h apparent double points, will besides intersect infjtvp—m(/i + v+p—2)+m(m—1)—2/i points; viz., the curve may,besides the apparent double points, have actual double points andstationary points; but these do not affect the formula.

80. Some caution is necessary in the application of the theorem. Forinstance, to consider cases that will present themselves in the sequel:let tho surfaces be cubics (fi = v=p=3) ; the number of remainingintersections is given as = 27— 7m+m (m—1)—2h. Suppose thatthe curve consists of four non-intersecting lines, ra=4, /i=6, thenumber is given as = —1. But observe in this case there are twolines each meeting the four given lines; that is, any cubic surfacepassing through the four given lines meets these two lines each ofthem in four points, that is, the cubic passes also through each of thetwo lines; the complete c^rve-intersection of the surfaces is made upof tho six lines m=6, h=7 (since each of the two lines, as intersectingthe four lines, gives actual double points, but the two lines togethergive one apparent double point), and the expression for the number ofthe remaining points of intersection becomes = 27—42 + 30—14 = 1,which is correct.

81. Similarly, if the given curve of intersection be a conic and twonon-intersecting lines, there is here in the plane of the conic a linemeeting each of the two given lines, and therefore meeting the cubicsurface, in four points, that is, lying wholly in the cubic surface : thecomplete cwrwe-intersection consists of the conic, the two given lines,and the last mentioned line m=5, 7t=5, and the number of points ofintersection is = 27—35 + 20—10= 2, which is correct. Again, ifthe given curve of intersection be two conies, here the line of intersec-tion of the planes of tho conies lies in the cubic surface; or, for thecomplete c?tm?-intersection we have ra=5, 7i=4; and the number ofpoints is 27—35 + 20 — 8 = 4. If in this last case each or either ofthe conies become a pair of intersecting lines, or if in the precedingcaso the conic becomes a pair of intersecting lines, the results remainunaltered.


82. If a surface of the order ft pass through a curve of the order mand class r without stationary points or actual double points, this im-poses on the surface a number of conditions = (/tz + l)ra—\r. In thecase in question, the value of r is = w(ra— l)—2ft; or, substituting,the number of conditions is = (f* + l ) m—\m (m—1) + h; and the for-mula in this form holds good even in the case where the curve hasstationary points and actual double points. Thus /u=3, the number ofconditions is = 4m—\m(m — l) + h. If the curve be a line, m = l ,h = 0, number of conditions is = 4 ; if the curve be a. pair of non-intersecting lines, m—2, %=1, number of conditions is = 8 . And soin general, if the curve consist of k non-intersecting lines (k = 4 atmost), then m = k, h = ^Jc(Jc—1), and the number of conditions is= 4&. If the curve be a conio, or a pair of intersecting lines, in = 2,h = 1, and the number of conditions is = 7. If the curve consist of klines, such that there are 0 pairs of intersecting lines, then m = k,h=.\k(k—1)—0, and the number of conditions is = 4&—0. It isobvious that, the number of conditions for a line being = 4, that for thek lines with 0 intersecting pairs must have the foregoing value 4&—0.In fact, when the lines do not intersect, we take on each line 4 points,and the cubic surface passing through any such 4 points will containthe line; but for two lines which intersect, taking this point, and oneach of the intei*secting lines 3 other points, the cubic surface throughthe 7 points will pass through the two lines; and so in other cases.

83. The formula must, in some instances, be applied with caution.Thus, given five non-intersecting lines &=5, 0=0, and the number ofconditions is = 20 ; and a cubic surface cannot be, in general, made topass through the lines. But if the five lines are met by any other line,then a cubic, surface, if it pass through the five lines, will pass throughthis sixth line; for the six lines k = 6, 0 = 5, and the number of con-ditions is 24—5 = 19; so that there is a determinate cubic surfacethrough the six lines, and therefore through the five lines related inthe manner just referred to.

84. Recurring to the problem of transformation, it appears by whatprecedes, that if the principal system in the first plane consists of at

points, and of a curve of the order wi, with hi apparent double points(the a, points being simple points, and the curve a simple curve on thesurfaces), then the conditions for a transformation are

= n8—1,

where, in the second line, instead of 3> I have written = . I remark,in passing, that I have ascertained that an actual triple point countsas an apparent double point; or, what is the same thing, that if the


curve has ^ actual triple points, then we may, instead of hit writehk + tx. The equations give

m, (4»—5—m,) = £ ( n - 1 ) (5nJ— n - 1 2 ) ~ 2 ^ ,( n - 4 ) wii-a = £ ( n - 1 ) (2n8-4n-15),

to which may be joined(3?i+8) m1-2m1 (w,-l)+4ft, + 5aj = ( n - 1 ) (6w + 17).

The first two equations for the successive values of n give

n = 2T, m, (8—»»i) = 2-2ft,, 2mi + cti = 5 ;n = 3, mi ( 7—mi) = 20—2/i,, mx+ai = 6;w = 4, w»i (11—7??|) = 64—2ftb <*i = — 1;n = 5, m, (15-mi) = 144-2ft,, -mi + ai =— 20 ;w = 6, w,(19-mi) = 270—2/i,, -2m1+«i = - 5 5 ;

&o. . &o. • &c.

' 85. It is remarkable that for n=4 there is no solution of the geo-metrical problem; in fact, ax = — 1, a negative value of a,, shows thatthis is so. For the higher values of n, there seem to be solutions withlarge values of mu ft,, a,; for example, w=5, we have mt = 20 +a,, is=20 at least. Writing m,=20, we have -100=144-2ft.,, or 2/t, = 244.The highest value of 2ht is = (TO,—1) (m,—2), which for m^ = 20 is=342; or the foregoing value 2ft, = 244 is admissible. Thus r»i = 20,ft., = 122, a, = 0 gives a solution; and, moreover, any larger valueof m,, say rrii = 20 +a, gives an admissible solution, vnx = 20+a,hi = 122+£a (a+ 25), ai=za. And so for n = 6, &c.; but I have notfurther examined any of these cases, and do not understand them.

There remain the cases n=2, n=3 . For n=2, since 2?M1 + a| = 5,we have wi, = 0, 1, or 2 ; tf», = 0 gives fti = 0, which is not admissible.The remaining solutions are m.x=:l, 7i,=0, a 1 =3 ; and m,=2, /J . ,^0 ,o, = l .

For n = 3, since w*, + a, = 6, we have mi = 0, 1, 2, 3, 4, 5, or 6.mi=0 gives ft,=10; m ,= l gives A,= 7; m,=2 gives fti=5; mx=zSgives A,=4 : these values are not geometrically admissible. The re-maining cases are m1=4, ft,=4, a,s=2; mi=5, ft,=5, a j = l ; ml=.Q1

7i,=7, «,=0.

86. The reciprocal transformation is in every case of the ordern '= n*—?ni. Hence considering the quadric transformations—

First, the case »> = 2, m, = l, ft, = 0, a, = 3 ; the reciprocal trans-formation is of the order n = 3. Suppose for a moment that the prin-cipal system in the second space is of the same nature as that aboveconsidered in the first space, consisting of a', points, and a curve of theorder m\ with h\ apparent double points (the a', points each a simplepoint, and the curve a simple curve on the surfaces X '= 0, &c.).


Passing back to the original transformation, we should have 2 = 9—wii,that is,. «&i = 7. But it has just been Been that, for n = 3 , the onlyvalues of mx are 4, 5, 6; hence for w'= 3 we cannot have mj=7. Theexplanation is, that the principal system in the second space is not ofthe form in question ; it, in fact, consists (as will appear) of three lineseach a simple line, and of another line which is a double line on thesurfaces X '= 0, &c. In the intersection of any two of these surfaces,the three lines count each once, the double line four times, and theorder of the curve of intersection is thus 3 + 4 = 7 , as it should be.The principal system may be characterized ai=0, «ii=3, 7ii=3, m'2=l,7 = 0.

Next, the case w=2, ^ = 2 , 7^=0, o i = l : the reciprocal transforma-tion is of the order w'= 2; it is evidently not of the form above considered(for this would make the original transformation to be of the order 3).Hence, assuming (as it seems allowable to do) that the principal Bystemdoes not contain any multiple point or curve, the reciprocal transfor-mation will be of the same form as the original one; viz., we shall havew'=2, w'i=2, &i=0, a i = l .

87. Considering next the cubic transformations, or those belonging ton = 3 ; in the case m1=4, 7^=4, 0^=2, the reciprocal transformation is otthe order 9 — 4 = 5; and in the case r^,=5, 7ii=5, ^ = 1 , the reciprocaltransformation is of the order 9—5 = 4 : I do not consider these cases.But mx = 6, 7ii = 7, ax = 0, the reciprocal transformation is of the order9—6 = 3 ; and assuming (as seems allowable) that the principalsystem does not contain any multiple point or curve, it must be of thesame form as the original transformation, that is, we must havew'=3, ra', = 6, 7ii=7, a\ = 0.

88. The transformations to be studied are thus,—1° The quadri-quadric transformation n=2, w I =2 , 7&i = 0, a, = l, and n '=2 , m\=21

7<j= 0, «*i=l; the principal system in each space consists of a point andof a conic (which may be a pair of intersecting lines) ; and the surfacesare quadrics. 2° The quadri-cubic transformation w = 2, w&i=l, 7ti=0,a!=3, and 7i'=3, aJ=O, ?%=3, h\ = 3, ?n^=l, 7^=0: in the firstspace the principal system consists of three points and a line, and the sur-faces are quadrics : in the second figure, the principal system consists ofthree simple lines and a double line; and the surfaces are cubic surfacespassing through this principal system, that is, they are cubic scrolls.3° The cubo-cubic transformation «=3 , «i=0, »ii=6, 7t,=7, andM ' = 3 , a i=0, wi,=6, Ai=7; in each space the principal system is aeextic curve with seven apparent double points (but there are differentcases to be considered according as the sextic curve does or does notbreak up into inferior curves), and the surfaces are cubic surfacesthrough the sextic curve.


Th$ Quadri-quadric Transformation between Two Spaces.

89. Starting from the equations x : y': %': vf — X : Y : Z : W, wehave here X=0 , &c, qnadric surfaces passing through a given pointand a given conic (which may be a pair of intersecting lines). Takeas = O, y = 0, 2 = 0 for the coordinates of the given point;' to=0 for theequation of the plane of the conic; the conic is then given as the in-tersection of this plane by a cone having the given point for its vertex;or say the equations of the conic arew = 0, (a, ...$as,y,&)a = 0; thegeneral equation of a quadric through the point and conic isw(ax+Py + yz)+8 (a, ...][a5,y,»)* = 0 j and it hence appears that theequations of the transformation may be taken to be

»': y': z': w'=xw : yw : zw : (a, ...][«, y, a)1;

these give at once a reciprocal system of the same form; viz., the two

sets are z: y: z' \ w'=xw : yw : zw : (a,...][», y, z)*,

and a?: y : z : w = x'w : y'vf: z'w' i {a,... ]£»',/, z')\

90. The Jacobian of the first space is at once found to be

that of the second space is of coursew*(a,...Xz',y',z'y = 0.

The two spaces are similar to each other; we may say that there is ineach of them a principal point and a principal conic; that the plane ofthe conic is the principal plane, and the cone having its vertex at thepoint and passing through the conic is the principal cone. To theprincipal point of either space corresponds any point whatever in theprincipal plane of the other space; and conversely. More definitely,the points of the one principal plane and the infinitesimal elements ofdirection through the principal point of the other space correspond ac-cording to the equation x : y : z = x : y : z. To any point on the prin-cipal conic of either space corresponds in the other space, not a mereelement of direction through the principal point of the other space, buta line of the principal cone; that is, to the points of the principal conicof the one space correspond the lines of the principal cone of the otherspace. The Jacobian of either space, consisting of the principal planetwice, and of the principal cone, is thus the principal counter-system ofthe other space.

91. [Writing (a, ...]lx,y,z)2 = ar'+i/'+z8, w = w'=l, the equationsof transformation become

x : y': z': 1 = x : y : z : xi+y*+zl,and * : y : z : 1 = x : y : z : x'--^-y'2 + z'2

t


or, what is the same thing, if for shortness

the equations arei x , y , z , x'

a 5 = ^ = £ 2 = 5 and SB =whence also r r ' = l ; this is the well known transformation by reci-procal radius vectors.]

92. The principal conic may be a pair of intersecting lines; takingits equations to be w=0, scy=O, the equations of transformation herebecome x': y': z : w' — xw : yw : zw : ay, .and x : y : z : w = asW: yW: z'vf: as'y'.There is no difficulty in the further development of the theory.

The Quadri-culic Transformation between Two Spaces. .

93. It will be convenient to have the unaccented letters (x, y, z, to)referring to the cubic surfaces. I will therefore take the quadric sur-faces in the second figure; viz., I will start from the equationsx : y t z : w = X': Y': Z': W , where X '=0 , r = 0 , Z '=0, W ' = 0 arequadric surfaces passing through three fixed points (say the principalpoints) and through a fixed line (say the principal line) in the secondfigure. Taking aj'=O, y '=0 for the planes passing through the prin-cipal line and through two of the principal points respectively ; «'= 0for the plane passing through the three principal points, w'= 0 for anarbitrary plane passing through the first mentioned two principal points,the implicit factors of x\ y, w may be so determined that for the thirdprincipal point x'=y'= —vf. That is, we shall have

for principal line as'= 0, y '= 0,for principal points (05'= 0, s '=0, w'=0),

'(/=0, /=0, «;'=0),» (x'=y=-wf

> ^=0),and this being so, the equation of a quadric surface through the prin-cipal points and line will be

(«x+Py')z'+yx'(y +«/) + ¥(»'+«>')»and the equations of transformation may be taken to be

x : y : z : w = x'z r yz : a?'(y'+ w') : y'(x'+ to').

94. Writing these in the extended formx:y:z:w:x—y: z—w =

x'z : yz': aj'(y' + tt>/) : z'(x' + w') : z'(x'—y) 1 w'(x'—y)and forming also the equation

xy : (xw—yz) = z': x'—y,


we at once derive the reciprocal system of equationsx : y': z : «/ = x (xw—yz) : y (xw—yz) : (x—y) xy : (z—to) xy,

so that this is a cubic transformation. And the cubic surface in thefirst space (corresponding to an arbitrary plane ax' '+ by+ cz+ did'== 0of the second space) is (ax+by) (xw —yz) + c(x—y)xy + d(z-w)xy = Q;viz., this is a cubic surface having the fixed double line (aj = O, y = 0),the fixed simple lines (aj = 0, z = 0), (y = 0, w = 0), and (x—y=0,z—w=0) ; it has also the variable simple line (dz+cx = 0, cZw + cy = 0).The principal figure of the first space thus consists of the thi-ee simplelines (05=0, «=0), (y=0, w=0), (as— y = 0, »—w = 0), and of the line(aj=O, y=0), a double line counting four times in the intersection oftwo of the cubic surfaces.

95. The cubic surface as having the double line (x=0, y=Q) is acubic scroll, and this line is the nodal directrix thereof; the line(dz+cx = 0, dw + cy = 0) is the simple directrix; the lines (aj=O, 2=0,),(y=0, w=.Q), (x—y=0, z—w=0) are at once seen to be lines meetinge.ach of these directrix lines; and they are generating lines of thescroll. To explain the generation of the scroll, observe that the sectionby any plane is a cubic curve having a given double point (viz., theintersection of the plane with the nodal directrix); and three other givenpoints (viz., the intersections of the plane with the three generating linesrespectively) ; this cubic also passes through the intersection of thoplane with the simple directrix. Conversely, if the plane be assumedat pleasure, and if, taking for the simple directrixany line which meotsthe given generating lines, we draw a cubic as above, then the scroll isthe scroll generated by a line which meets each of the directrix lines,and also the cubic.

If the plane be taken to pass through any generating line, then thecubic section breaks up into this line, and a conic; the conic does notmeet the (simple directrix, but it meets the nodal directrix; and anysuch conic will serve as a directrix; viz., the scroll is generated by thelines which meet the two directrix lines and the conic.

96. Any two scrolls as above meet in the three fixed generatinglines, and in the nodal directrix counting four times; they consequentlymeet besides in a curve of the second order, which is a conic (one of theconies just referred to). In order to further explain the theoiy, sup-pose for a moment that the two scrolls had only a common nodal di-rectrix ; they would besides meet in a quintic curve; this curve wouldmeet the nodal directrix in four points, viz., tho points ut whichthe two scrolls have a common tangent plane. Now if at any point ofthe nodal directrix the two scrolls have a common generating line, thentho plane through this lino and the nodal line is one of the two tangent

Rational Transformation bettveen Two Spaces. 173

planes of each scroll; that is, the scrolls have this plane for a commontangent plane. Hence, in the case of the common three generatinglines, the points where these meet the nodal line are three ofthe four points just referred to; there remains therefore one point,which is the point wKere the conic meets the nodal line; through thispoint, there are for each of the scrolls two generating lines; one ofthese for the first scroll, and one for the second scroll, lie in a planewith the nodal line; the other two determine the plane of the conic;and the tangent to the conic at its intersection with the nodal line isthe intersection of the plane of the conic with the plane of the first-mentioned two generating lines.

97. Analytically we have the two equations

c (x—y) xy+(ax + by) (xw—yz) + d(z—w) xy = 0,c (x —y) xy+(ax+b'y)(xw —yz) +d'(z—w)xy = 0;

or, combining these equations so as to eliminate successively the termsin x (xw—yz) and y (xw—yz), and for this purpose writing(bo'—b'c, ca'—ca, ab'—a'b, ad'—a'd, bd'—b'd, cd'—cd) = (a,b,c,f, g,h),

and therefore af+bg+ch = 0,we have b (x—y) x—c (xw—yz)—f (z—w) x = 0,

—a (as—y)y + c(xw—yz)—g(z—w)y = 0,

and multiplying the first of these by c+g and the second by c—f, andadding, the whole divides by x —y, and the final result is

(c + g ) (b* - f 2 ) - ( c - f ) ( ay+g« ; ) = 0;

viz., this is the equation of the plane of the conic.

98. Any two scrolls as above meeting in a conic, a third scroll willmeet the conic in six points; but these include the point on the nodaldirectrix twice, and the points on the three fixed generating lines eachonce; there is left a single point of intersection, viz., this is the onevariable point of intersection of the three scrolls; which is in accord-ance with the theory.

99. For the Jacobian of the second space, we have

z , 0 , x, 0 = 0 ;0 , z' , y\ 0

y'+w, x , 0, 0y , x'+w, 0, y

that is, 2x'yz'(x'— y) = 0; viz., z'=0 is the plane containing thethree principal points; and aj'=O, y'=0, x'—y'=0 are the planeswhich pass through the principal line and the three principal pointsrespectively.

174 Prof. Cay ley on the100. For the Jacobian of the first space, we have

2xw—yz, —xz , — xy, x2

yw , xw — 2yz, —if, xy2xy—yi

1 x*—2xy, 0 , P(z—w)y, (z—w)x, xy, —xy

= 0;

that is, 3asV (»—i/Y (xu>—yz) — 0 ; viz., a? = 0, y = 0, as—y = 0 arethe planes through the nodal directiix and the three fixed generatorsrespectively (each plane therefore occurring twice); and ano—yz = 0is the qnadric scroll generated by the lines which meet each of thethree generators («=0, z=0), (y—0, w=0), (»—y=0, z—u> = 0 ) ;this scroll passing also through the nodal directrix 3=0, y=0.

The Oubo-cubic Transformation between Two Spaces.

101. The principal system in the first space is a sextic curve with 7apparent double points; but this curve may be either a single curve,or it may break up into inferior curves. I have not examined all thecases which may arise; but the two extreme cases are—(A) The sexticcurve breaks up into six lines, viz., two non-intersecting lines, and fourother lines each meeting each of the two lines (this implies that no twoof the four lines meet each other) : here the two lines give 1 apparentdouble point, and the four lines give 6 apparent double points; totalnumber is = 7, as it should be. (B) The curve is a proper sextiocurve, with 7 apparent double points : this gives, as will be shown,the general lineo-linear transformation. The two cases are each ofthem symmetrical.

(A) The Principal System consists of Six Lines.

102. Taking in the first space, for the equations of the two lines,(a>=0, y=O) and (z=0, w=0), and for the equations of the four lines,(a>=0, e=0), (y=0,w=0), (x-y=0, z-w=Q), (x-py=0, z-qw-0),then, if the equations of transformation are taken to be

x'—py :x'—y : z — qw : z'—w = (,^—py) ( xw— yz)

: (z — qw) ( xw— yz):{z— w) (qxw—pyz) ;

these lead conversely (see post, No. 104) to a like system,

x—py : x—y : z—qio : z—w = (x'—py) M': ( a / - y')K: (z'— qw) M': (z- w') N',


where for shortnessM' = p (q — l)2xw'— q (p—l)2yV+ (pq— 1) (p —q)y'iv,N'= ( 2 - l ) W - (p-iyyz'+ (pq-r)(p

or, as these are better written,

M' = — q(p-l)y'(p — lz — pq — lw) +p(q—l)io'(q—lx'—pq-lyf),N' = - (p—l)y'(p — lz — pq-lw) + (q—l)w'(q-lz— pq—ly).

Hence the principal system in the second plane is composed of thetwo non-intersecting lines (a?'=0, y ' = 0 ) , (z '= 0, w '=0) and thefour lines (p—lz'—pq—lw'= 0, g—la/—pg—ly'= 0), (y '=0, w'=0),(a;'—y'= 0, z'—w/= 0), (a?'—py'= 0, z'—qw'=z 0), each meeting eachof the two lines.

103. The Jacobian of the first space is

yz—pywi —xz—panv+2pyzt —y(x—py) , x(x—py) = 0,Zqscw—pyz—qyw, —pxz—qxw + 2pyz, —py(x—y) , qx(as—y)

w(z—qw) , —z(z—qw) , oew—2yz + qyw, xz—2qxw + qyzqw (z—w) , —pz (z—to) , qxw—2pyz+pyw, qxz—2qocw+pyz

viz., this is xyzw (x—y) (z—py) (z—w) (z—qw) = 0, the equation ofthe planes each passing through one of the four lines and one of thetwo lines.

Similarly, the Jacobian of the second space is

(q— 1 oo —pq—1 y) (p—lz'—pq—lw) (x—y) (z — w)X (x'—py) (z- qw)y'w = 0 ;

viz., this is the equation of the eight planes each passing through oneof the four lines and one of the two lines.

104. To effect the foregoing transformation, writing

%': y : z*: w = (n—py) ( xw — yz):(x— y) (qxw-pyz): (a — qw) ( xw— yz): (z— w) (qxw—pyz) ;

or what will ultimately be the same thing, but it is more convenientfor working with,

• x = (x—py) ( xw— yz),.y — (®- y)(qxw-pyz),z = (z-qw) ( xw— yz),

w' = (z— w) (qxw—pyz);these give x —py = M V,

»— y — N Y»z—qio=. MV,z— w = N'w',


where M', N' are quantities which have to be determined; and thence'

(1— p) z = M'x'—p'N'y,

(1—q) z = MV — qN'w',(L—q)w = MV — N'u/;

whence also(1— p)(l-q)( xw— yz) — N'{(2—lx'w'—p—ly'z')M'+p—qy'wT$'\,

(1 -_p) (1 - g) (qxw-pyz) =M'{ — (p — q) x'z~bA!+ (pq—q x'w'—pq—p i/V)N'};

but we havexw—yz _ x __ x—py _ x' _ MV _ N_' #

qlcw^pyz y ' x—y ~ y' ' Ky ~ M''

or substituting

M' (q—1 »V —jp—1 yV) + N' (jp — q) y'w

= M' (—p —qx'z') + m'(pq — q x'w'—pq—py'z) ;

that is M'{q—1 xw' —p—1 yV-\-p—q xz J

= W\pq-~qx'w—pq—py'z'-v-qy'w'\;or, what is the same thing,

M' = pq—q xw'—pq—p y'z'—p—qy'w,

N' = 2—1 xw'— p—ly'z'+p—q xz';

viz., M', N' having these values, the original equations

x : y : z': w' = (x —py) ( xw— yz): (a?— y ) (paMW — 2 y a )

: (z — qw) ( a?M>— y«)

: (z— «>) (j?a!W—qyz),

give to—py - x—y : z—gw : as—to = MV: N'y': MV:

If, in these equations, in place of (#', y\ z, w') we write Qf'—py', x' -y\%—qw\ z—w'), the new values of M', N' are found to be

W =p (j—1) x'to—q (p—iy y'z'+ (pq — 1) (p—q) y'w,N' = (2—I)2 a»V— (p—I)2 y'z + (pq —A) (p—q) yw,

and we have the formulae of No. 102.

(B) The Principal System of a Proper Sextic Curve ; the Lineo-linear Transformation between Two Spaces.

105. I start with the lineo-linear transformation, and show that thisis in fact a transformation such that the principal system in eitherspace is a sextic curve with seven apparent double points. I do not

Rational Transformation between Two Spaces. Ill

attempt any formal proof, but assume that the lineo-linear transforma-tion is the most general one which gives rise to such a principal system.

We have between (a?, y, z, 10), (of, y, z', w') three lineo-linear equations;writing these first under the form

(P,, Qb R,, S,£*', y\ z\ w) = 0,(P2, Q2, &>, S2$>', y, z, w) — 0,(P3, Q3, R3, S3$>', y, z, w) = 0,

we have x : y : z': w = X : Y : Z : W, where X, Y, Z, W are the deter-minants (each with its proper sign) formed out of the matrix

1*3) S3

106. Each of the surfaces X=0, Y=0, Z=0, W = 0 , or generallyany surface aX+&Y+cZ + cZW = 0, is thus a cubic surface passingthrough the curve . Pi, Qb Ri, Sj = 0,

2) ^52) -"2 , ° S

P3, Q3, R3, S3

which is at otfce seen to be of the order 6. In fact any two of thesesurfaces, for instance

Px, Q,, = 0 and Pi, Qi, S,P2, Q2> S 2

1*3) Q3> S3

= 0,

3> **3i R 3

have in common a curveH p p p _ n

rU * 2> -^3 — v ,

II Q b Q2> Q3which is of the order 3 ; they consequently besides intersect in a curveof the order 6, which is the before mentioned curve of intersection ofall the surfaces. And it further appears that the number of the ap-parent double points is = 7; in fact the formula in the case of twosurfaces of the orders p, v, the complete intersection of which consists ofa curve of the order m with h apparent double points, and of a curveof the order m with h' apparent double points, the numbers m, vi, h, h'are connected by the equation 2 (h—li) = (m—in) (ft—1) (v — 1).(Salmon's Solid Geom., 2nd Ed., p. 273.) Hence, in the case of thetwo cubic suxfaces intersecting as above (since for the cubic curve wehave m'= 3, 7&'= 1, and for the sextic m = 6), the formula becomes2 (h—1) = 12, that is h=l + 6 = 7; or the number of apparent doublepoints is = 7.

107. It thus appears that the principal system in the first plane is acurve of the order 6, with seven apparent double points: it is to beadded that there are not in general any actual double points or sta-

N


tionary points, so that the class of the curve is 6.5—2.7, =16 , andits deficiency is \ 5 .4— 7, = 3. For convenience I will refer to this asthe curve 2.

The transformation is obviously a symmetrical one; hence the prin-cipal system in the second space is in like manner a curve of the order6, with seven apparent double points; say it is the curve 2'.

108. Consider in the first space any point P on the curve 2 ; for thispoint the three equations

(P,, Q,, R,, Sj^Y, y\ z, w') = 0,(P2, Q,, R,, S2£ „ ) = 0,(P8, Q8, B* S,][ „ ) = 0,

are not independent, but are equivalent to two linear equations in(x,y\ z, w); that is, to the point P on the curve 2 there corresponds inthe second space, not a determinate point P', but any point whatever on acertain line L'; or say to the point P on 2 there corresponds a line I/ ;and as P describes the curve 2, L' describes a scroll IT; that is, to thecurve 2 there corresponds a scroll IT, the principal counter-system inthe second space. Similarly to the curve 2' there corresponds a scrollIT, the principal counter-system of the first space.

109. The scroll II is the Jacobian of the first space; and as such it isof the order 8, having the curve 2 for a triple line—and it thusappears that the Jacobian of the first space is a scroll (a theorem theanalytical verification of which seems by no means easy). But withoutassuming the identity of the scroll IT with this Jacobian, or taking theorder of the scroll to be known, I proceed to show that the scroll II isthe scroll generated by the lines each of which meets the curve 2 threetimes ; it will thereby appear that the order is = 8, and that the curveis a triple line on the scroll.

Consider a point P'on 2', and the corresponding line L of the first space:take 6' a plane in the second place; corresponding to it the cubic surface6 in the first space. By imposing a single relation on the coefficients(a, b, c, d) in the equation ax + by + cz -f dw = 0 of the plane 6', wemake it pass through the point P*; therefore by imposing this same

' single relation on the coefficients (a, b, c~, d) of the cubic surface 9, wemake it pass through the line L ; 0 is a cubic surface through 2 ; andit is easy to see that the effect will be as above only if the line L cutsthe curve 2 three times ; this being so, the general cubic surface 6meets L in three points (viz., the three intersections of L with 2), andif 9 be made to pass through a fourth point on the line L, it will passthrough the line L; it thus appears that the line L meets 2 three times,and consequently that the scroll IT is generated by the lines whichmeet 2 three times.


110. The theory of a scroll so generated is considered in my "Memoiron Skew surfaces, otherwise Scrolls."* Writing m=6, h=7 andtherefore M[=— £m(ra—l) + 7i]=— 8, the order of the scroll is( i M 3 + ( m ~ 2 ) M=40—32) = 8; but calculating the values of

NG(ms) = }[m]4 + 6m+M (3[m]2-12m+33) + M2. 3,

NR (ms) = TV[m]6+f[m]5- |[m]3-3m+M (A[m]4-£[m]3-fm2 + 8m-20) +M2 ( | [m]2-2m);

these are found to be respectively = 0; viz., there are no nodalgenerators, and no nodal residue; the sextic curve 2 is a triple curveon the surface, and there is not any other multiple line.

111. It may be remarked that any plane 9' meets the sextic curve 2' insix points; hence the corresponding cubic surface 6 contains six lines,generatrices of IT, and, therefore, each meeting the curve 2 threetimes ; say six lines L. Through one of these lines L, draw to thecubic surface 0 a triple tangent plane meeting it in the line L and intwo other lines, say M, N ; this plane must meet 2 in three new pointswhich must lie on the lines M, N ; viz., one of these lines must passthrough two of the points, and the other line through the third point.

Addition—September, 1870.

The formulro of No. 84 are included in the following more generalformulae ; viz., if the principal system consist of a, points, each a simplepoint, a2 points each a quadri-conical point, a8 points each a cubi-conicalpoint, &c, and of a simple curve order mt with hi apparent doublepoints, a double curve order ma with h, apparent double points, and soon; and if moreover, the curves mu m^ intersect in klti points, the curvesmu m^ in &13 points, &c.; then writing in general p = \m (m—l)—h;that is, pi = tyiii (mi—1)— hi, p2 = %mi(m2—l) — li2, &c, I find thatthe general condition of equivalence is

ai + (Sn—l)mx —2px

+ 8o2 + (12?i —16) ma—16p2

(Srhi - 2^) mr—2r3pr

- 2 8 ^ , 3

...s\Sr-s)k,ir(s<r)

and that the general condition of postulation is

* Phil. Trans., vol. cliii. 1863, pp. 453—483. See the Table S(m3) &c, p.457 ; inthe value of NR (>n8) instead of term + 3»» read — 3w.

180 On the Bational Transformation between Two Sjpaces.

(r + 1) n— -5)]

>- 8 * , , ,- 7 s («+1X'+ l—tf+2) A., r (« < r)

in whioh formulee it is however assumed that the curves have not anyactual multiple points. This implies that if any one of the curves, saymn break up into two or more curves, the component curves do notintersect each other; for, of course, any such point of intersectionwould be an actual double point on the curve mr. I. believe, however,that the formulae will extend to this case by admitting for a the value«=r ; viz., if we suppose the curve mr to be the aggregate of the twocurves m'r, m" intersecting in K, points, then that the correspondingterms in the equivalence-equation are

and that those in the postulation-equation are

Let the r-tuple curve consist of three right lines meeting in a point:this is an actual triple point, and the formulas do not apply. Butcalculating the postulation*terms by the formula, we have mr = 3,pr = £3.2—0, =3 j and the terms are

- i [(»—!)(»— 2)(r-S)(r-4)+4r(r+l)(2r+l)] ,which are = | r ( r+ l ) (3? i -4 r+6) - | ( r - l ) ( r -2 ) ( r -3 ) ( r -4 ) ,or say = |r(r+l)(3»—4r+e) + |(—Z+lOr8—SW + SOr—24).I have found by an independent investigation that this value requires

the correction +\ (^-8r3 + 30ra-56r+24+ hzizi)ll

and that the true value of the postulation is

\

viz., that this is the number of the conditions to be satisfied that a sur-face of the order n may have for an r-tuple curve three given rightlines meeting in a point.

1870.] On the Cartesian with Two Imaginary Axial Foot. 181

June 9th, 1870.

Professor CAYLBY, President, in the Chair.The Hon. Sir James Cockle, F.R.S., was elected a Member.The President read the following

Note on the Cartesian with Two Imaginary Axial Foci.

Let A, A', B, B' be a pair of points and anti-points;viz., (A, A') the two imaginary points, coordinates (±jSt, 0),

(B, B') the two real points, coordinates (0, db (3) ;and write p, p', <r, </ for the distances of a point (x, y) from the fourpoints respectively; say

9 "

We have pa+p'a = 2aja + 2ya-2/3a = fc+a1"—4/39,pp '=

and thence (p+p ' ) 9 = (<r+o'f -

or say

The equation of a Cartesian having the two imaginary axial foci A, A' is

viz., this is p (p+p') + gi(p —p') + 2fca = 0;

or, what is the same thing, it is= 0,

which is the equation expressed in terms of the distances <r, o' from thenon-axial real foci Bj B'. Of course, the radicals are to be taken withthe signs ± . This equation gives, however, the Cartesian in combi-nation with an equal curve situate symmetrically therewith in regardto the axis of y.

The distances <r, a may conveniently be expressed in terms of asingle variable parameter 0; in fact, we may write

'y—44& = — M—lcd,?- ( » - o'y =s - fca + led;

that is. ( + O 8 4 / 3 a = 9

VOL. HI.—NO. 32.

182 On the Cartesian with Two Imaginary Axial Foci. [June 9,

and therefore <*+</= A / 3 2 + ^v p

so that, assigning to 6 any given value, we have <r, a\ and thence theposition of the point on the curve. "We may draw the hyperbola

if = 409 + — »8, and the ellipse y* = 4/3a j-aj2; and then measuring

oflf in these two curves respectively the ordinates which belong to theabscissro £ + 0 for the hyperbola, k—6 for the ellipse, we have the values<r + ff' and cr—a, which determine the point on the curve. Consideringk>P> ?>$ as disposable quantities, the conies may be any ellipse andhyperbola whatever, having a pair of vertices in common; and thecomplete construction is,—From thefixedpointK in the axis of x, measure oflf in opposite direc-tions the equal distances KM, KN, and take

<r + a the ordinate at M in the hyperbola,±(<r—a) „ „ N „ ellipse;where a, a denote the distances of the re-quired point from the fixed points B and B'respectively, the distance of each of these fromthe origin being = £ the common semi-axis.We may imagine N travelling from one extremity of the z-axis of theellipse to the other, the value of a + a will be real and greater than BB',that oia — o real and less than BB', and the point (o-, </) will be real.The construction gives, it will be observed, the two symmetricallysituated curves.

The aj-semi-axis of the ellipse is *• 2/3, and the form of the curve de-K

pends chiefly on the value of the ratio h : ^ 2/3 ; or what is the same

thing, k2: 2flg. We see, for instance, that, in order that the curve maymeet the axis of y in two real points between the foci, the value d=—k

must give a real value of a—<r; viz., that we must have 4/3* > —— ;

that is, /3y > ¥, or & < fa. If k has this value, viz., k = 112/3

= \ semi-axis, the curve touches the axis of y at the origin; ifk < \ semi-axis, the curve cuts the axis of y in two real points be-tween the foci; if k > £ semi-axis, the curve does not cut the axis of ybetween the foci.

Mr. Cotterill then gave an account of his paper " On the Intersectionof Plane Curves/' A short discussion ensued on the subject of thecommunication.

1870.] Proceedings. 183

' The following presents were made to the Society at this meeting andsubsequently:—

"Journal of the Institute of Actuaries," No. lxxix., for April, 1870." Monatsbericht," April, May, July, 1870." On Reciprocal Figures, Frames, and Diagrams of Forces," by J.

Clerk Maxwell, F.R.S. (Transactions of Royal Society of Edinburgh,vol. xxvi., 1870) ; and " On Governors," by the same (Proceedings ofRoyal Society, No. c , 1868) : from the Author.

" Transactions of Royal Irish Academy," vol. xxiv., pts. iv., viii.—xv.inclusive.

" Annali di Matematica," Serie iia, torn. iii°, fasc. 4°, May, 1870." Annuaire de l'Acad£inie Royale des Sciences, des Lettres, et des

Beaux-Arts de Belgique, 1870," and the " Bulletins" of the sameAcademy, 2e Serie, torn, xxvii. and torn, xxviii., 1869.

" Crelle," 71r Band (4s Heft) and 72r Band (I1, 2$ and 3* Heft)." Journal de l'Ecole Impe'riale Polytechnique, publie par le Conseil

d'Instruction," (43ecahier), torn, xxvi., 1870." On the Contact of Conies with Surfaoes," by W. Spottiswoode,

F.R.S.: from the Author.From the Smithsonian Institution, Washington, U.S.: the "Report"

for 1868; the " Orbit and Phsenomena of a Meteoric Fire-ball," byDr. J. H. Coffin; the " Transatlantic Longitude as determined by theCoast Survey Expedition of 1866," by B. Apthorp Gould.

" Sugli integrali a differenziale algebrico," Memoria di L. Cremona:from the Author.

"Quelques re"sultats obtenus par la consideration du defacementinfiniment petit d'une surface algebrique;" and "Determination duplan osculateur et du rayon de courbure de la trajectoire d'un pointquelconque d'une droite que Ton d6place en l'assujetissant a certaineaconditions," by M. A. Mannheim; from the Author,

02

ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES

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