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Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW) Citation: L.E.J. Brouwer, On a decomposition of a continuous motion about a fixed point O of S4 into twocontinuous motions about O of S3's, in:KNAW, Proceedings, 6, 1903-1904, Amsterdam, 1904, pp. 716-735 This PDF was made on 24 September 2010, from the 'Digital Library' of the Dutch History of Science Web Center (www.dwc.knaw.nl)

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Mathematics. - "On a clecompvsition of a continuOltS 'l1lvli,on aboztt a ji.'lJcd point 0 of 81 i1?to two continuous motions abo?lt 0 of 8a's" by MI'. L. E. .T. BROUWJ<!R, conl1nunicated by Prof. KORTEWBG.

(Communicated in the meeting of FebrualY 27, 1904).

Two planes in 84 making two equal angles of position are called 1l1utnally "eqniangnlal' to the right" if one is (with its n01'111a1 plane) plane of rotation for au equianglllal' double rotntion to the right about the other one and its normal plane.

W' e wiII eaU the sides of one alld the same acute angle of position "col'l'esponding vectors" thl'ough the point of intel'section of two equianguhtl' intersecting planes.

As is knowl1 a system of planes equiangl1lal' to the l'ight or to the Ieft is infinite of order two. Of COlll'se a detel'IUined eqniangulal' system of plan es to tbe right can have with a detcrmined eqllian­gular system of planes to tbe 1eft not more than ODe pair of planes in common (two pairs of plan es canllot illterscct each other at the same time equiangula1'1y to the right aml to the Ieft); but one pair of planes they al ways have in common. We will show how that common pair of planes eau be found.

A pair of intersectil1g pairs of planes of both systems is of course easy to find. We lay through any vector OC the planes belonging to the two systems; their nOl'mal plan es intel'sect each other in a second vector OD. Thus OCD is one plane of position of those two pairs of p1anes. In the second plane of POSitiOll the four planes furnish four lines of intel'section,-' let us say OH, OF, OJ(, OG, in such a way, that the considered pairs of planes must be OCH; o DJ( and o CF; 0 DG. The following figures are supposed to be situated in those two planes of position.

Sf C. I

AI

~---e r

,~..---A

Fig. 1. Let the pair of planes OCH; ODK belong to the ghren system

equianglllal' to the l'ight, and OCF; ODG t0 the given system equiangular to the 1eft.

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Tltc ycclors in t11e &eeond plane of position are chmvn iu snel! a Wt"ty that either

\ Oll ~ oe ... IOK~oD

(1)

is au equiangulal' double l'otation to the l'ight, Ol' t.hat sneh i& the case with

\OIJ~Oe

IOK~OD' ,Ve slla11 bllppOSC thc fh'st to uc trne (tbe l'ea&olling is tbe same for the &eeond ease). Thell

\OF~Oe

IOG~OD is also all equianglliar double l'otation to the l'ight; fol', the plancs OPC and OGn can be brought 10 C'oinclde with these directions of rotaHon with the plancs OllC alld OJ(D, ha ving the dil'ections of )'otation

So

\ OII ~ oe IOJ(~OD

!OF~ oe

. . . (2) OG'~OD

is an eqniangular double rotation to the left. If fhl'thel'mol'e OA and OB are biseetors of the angles HOP and

](OG, anel if we have made L COAt = L DOB' = LHOA = LPUA = LJ(OB = LGOB, then the pair of planes

\ A.OA' I 8 OB'

is a pair of planes of l'otation as weIl for the equiangular double rotation to the l'ight (1) as for the eqlliangnlal' double rotation to the left (2). 80 it is the pair of eommon planes whieh was looked for of the two systems of planes,

c'

A

Fig. 2.

drawn upwards lie in

IJ'

\Ve sha11 think now that through two arbitrary vee tors OA and OB two 1)lancs interseeting eaeh oUier eqniangnlarly to the left luwe becn laid; wc slwJl I10W consielel' more elosely the position whielt two sU0h planes have with respeet to the plane OAB and its 11or111al plane. 'V c sha11 eaU the inelicateel elluiang ular planes to the left Ct and Ij; and indicate OAB by r mul its n01'111al pIane by ó. In fig. 2 thc lines

ó and those drawn downwards in y.

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Thc plane DCC' is the pinne of position of rand a, intersectcd by r in OP, bJ' a in OC. Fig. 3 is supposed to lie in_ that phtne of position. We have made fal'theron in fig. 1 the t~ngles A' OB', C'OD', COD eqnal to LAOB=p, and l11e dil'ections of l'otittion indicated in those planes belong to a double rotatioll to the right. Fig. 4 is supposed to lie in tbe plane ODD', anel the lines OG and OG' are drawn in it in sueh a way, that ODG D' G' :=:. OeFC' F'.

c'

G

~""'""' ___ c -----D o Fig. 3. Fig. 4.

We shall eonsider the plane BOG more elosely. I.Jet us project' OB and OG both on (c, theu it is not difficult 10 see that the executioll of those two opel'ations, eaeh of whieb is thl'eedimeusionaI, gives as a l'esult two lines OR aud OK, mntually peL'pendienlal' (see fig, 5, supposed to lie in a).

The projecting plan es are successively: OBF' (fig. 6) and OGA' (fig. 7).

ioo::Ir------ K T' A'

01.00::::::::----- J{

Fig. 5. Fig. 6. Fig. 7

We sha11 directly see that OB aud OF' are situated on diffe­rent sides of OH, and OG aud OA' on different sides of OK, aud that LEOB = LKOG, ifwe suppose oUl'selves to be successively in the thl'eedimellsional spaces, in which the pro.jecting takes place.

80 we see that the plane BOG hns two mutually perpendicular vectors, ll1aking equal al1g1es with a and pl'oJecting itselt' on a

accol'dil1g to two perpendiculal' vectol'S nall1ely OB and OG, pl'ojecting themselves accol'ding to OH aud Of{: the chal'acteristie of equiangular intel'seclÎon.

I.Jet ns still exall1ine of which kind that equiangular intorsection is; we shall then pel'ceive that on account of OB being transferl'ed into OG by the equiangular double rotation

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OF' --'» OA' o II --'» 0 J(

and this being of the same kind as OF' --'» OA' 0.11--'» OF

which in its turn ean be made to eoineide with oe' --'» OA' OA --'» oe

by a single l'otation about the plane OAA', the kind of equiangular interseetion is the same as tlle kind of the double rotation

oe' --'» OA' OA --'» oe

"vhicb is to the lefL according to tlle data. So Ihe plane nBG is identical with the plane (j, for throngh OB ,

only one plane eqniangulal' to the left with a call pass.

If we now intl'oduce tlle notation "C~) equiangular to the right"

indicatillg if abeel denote foU!" vectort, through 0, that the planes (,tb) and (cd) are eqniangnlal' to the right and that the same double equiangnlal' l'otation to the right transfel'ring a into b, also transfers c into cl, then

(OA, OB) OF,OG

is equiangular to the right anel (he cOrl'esponding equiangulal' double l'otation to the right transfers « into (jo In other words we have pl'oYed tlle

Theorem 1. If G!) is equiangulal' to tlle right, then G;) is equian­

glllar to the left; or in other words though less significant: By an eqniangular double rota.tion to the l'ight any plane pa.sses

into one equiangular to it to the left. If we suppose three vectors abc (whose position of course determines

the pot,ition of 84) 10 have come aftel' some equiangular double rotations

to the l'ight into the position dej~ then (::) is equiangttlm' to tIle left

and (~;) equiangnlal' to the left; so G~) equiang'lllar to tIle l'ight

anel C~) equianglllal' to tJle right, so finally

(~;)

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equiangnlar to the right; in other words the final position would have been obtainable out of the initial positiol1 by a single equiangnlar double l'otation to the l'ight; witlr whieh is proven:

'Pheol'em 2. Equiangnlar double rotations of tbe same kind form a

group. Let us snppose given two eqniangular systems to the right and two

vectors OA anel OB throngh each of which we bring thè plan es belonging to both systems; then thc eqniangular double rotation to the left, tl'ansferring OA inio OB, wiII transfer at the same time the angle of position iormeu in OA into the o~e formed in OB, thus :

TAeo1'em 3. Two eqniangulm systems to the right form in each vector the same angle, which can be callcd the angle of the two systems.

The obtainecl resnlts we shall vel'ify by dedneïng analytically theorems 1 and 3, which deduction will also thl'oW some more light.

Snppose a reetanglliar system of coordinnteó to be given in sneh a way that

(OAS'OXS )

OX1 , 0x.4

is cquia,ngnhw to the right. The same thC11 hol<.1s good for

(OX" OXl) and (OX 1 , OX2).

OX2, 01.4 OX3; OA4

A vector a thl'ough 0 we ean determine by its foUl' cosines of direction al' a2, as, a4.

A plane pasóing thl'ough the vectors a and /1 with direction of rotation from a to {/, is determined by its six coeffieients of pOSiti011 (i. e' pl'ojections of a vector unity) 1. 23 , 1.31l ) u' ),14' )'24' )'3-4'

which are defined by the following l'elations, if we represent a2{/3-a3f12 by g23: c

ts t )'2~ = , -e c • . + VS2S

2 + g312 + 612

2 + 614 2 + 6./ + 63 /

We must take the positive sign in the denominatol', fol' ).., LuuSt be pO&itive, when the projection of a on OX.Xa to th at of 1~ on OX.Xa

l'otates throngh an angle less tban :Ir in the same way as OX2 to OXs ; and in that same case 62 3 gives us a positive value. If we now l'epl'eSent that positive l'OOt of the denominatol' by 1(, then

1 623 , S31 t 11.., = JC; "31 = K' e c.

An equiangulal' double l'ota.tion to tbe l'ight eau be given by tlle

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system of equiangular planes of rotation to the l'ight with direction and the angle of l'otation.

For all those planes of l'otation

.Î.23 +.Î.14

)·31 +.Î.H

.112 + ÄS4

have the same values. These three values al' a., as, besides tlle cosinus of the angle of rotation a 4' we ean take as determining quantities of the equiangular double rotation to the right. A rotation < 2:1r is unequi\'oeally determinecl by that (for, whether the angle

of rotation is ~:Ir, whieh is left undecided by the value of the cosinus,

follows from the sign of the al' a" as). Suppose an arbitrary vector a to be transfel'red by the rotation

into [:1, then it holds good for earl! pair of vectors a[:1 th at :

a2 [:1s - as [:1, + al [:14 - a4 [:11 = f( . al

as t~l - al [:13 + a. {J4 - a4 {J. = f( . a,

al {J. - a. [:11 + as {J4 - a4 {Js = K. as

al [:11 + a. [:1. + as {Js + a4 {J4 = a4 •

Ir however we considel' that J( = + Vsin 2 {t, if {t is the~angle of rotation, then J( proves to be a constant for all pairs of vectors so that "'Ie may regard 1(, al' J(. a" K. as and a4 as determining quan­tities of tbe double rotation which we sha11 caU :;rIl :Ir,. :;ra.:;r4; and we sh all write the relations:

- a 4 1~1 - as {J. + a. {Js + al [:14 =:lr1 ('

((s {Jl - a4 {J. - al {Js + a. {J4 = :;r. - a. {Jl + al [:1. - a4 fis + as {i4 = :;rs

al fll + a. fI. + as fis + ((4 {l4 = :lr4

. (H)

in which we have at the same time arranged the first members aceording to I~l' [:1 •• I~s. {j4. We now perceive:

:TrI' + 31:,' + :Ira' +:;r/ = [(2 !P'n' +.Î.81 ' + .Î.12• + )·14' + )·24' + ).3/) + + 2 (Î.n l.14 + 1. 31 )..4 +).12 ).34)} + a/.

= J{' (1 + 2 X 0) + a/ =sin' {t + cos' {t

=1.

So we can l'egard :;ril :71',. :l3':;r4 as C'osines of direction of a vector throngh 0 in 84 anel wc ean rCpl'eSellL all eqnianglllal' double rotation 10 thc right uy a vector thl'ough 0 in 84 , ,,·hieh detel'mines it

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unequivocally, (A vector without length; lateron we shalJ determine it, likewise uneqnivocally, uy a vector witlt length, in Sa)'

If we farthermore eonsider the determinant on the coefficients of {jl' {j2' I~" (J. in (H) it proves to Ratisfy all the conditions of an orthogonal transformation ,

Weeall that transformation with that determinant

- a4 - aa + a, + al

+ as -a _ 4 - al + a2

- a2 + al - a. +aa + al + a2 + as + a.

the (+ 1') a-tl'ansfol'mation; it appeal's in the relations (H), if the fil'&t membel's are arranged aecording to the cosines of direction of the fin al position of the rotating vector,_ If they are arranged according to the cosines of direction of the initial position the determinant of the coeffieients beeomes

{j4 {ja - {j2 - fJI

- fJa fJ. fJI - fJ2

{j2 - fJI fJ. -- fJa

{jl fJ2 fJa fJ. ,

which we shall eall the (-1') fJ-transformation, Qnite analogous to this we have for equiangular double rotations

to the left ({>I Q2 Qa (4) bilinear homogeneous equations between the cosines of direction of initial and final position of a vector, let us say a anel {j, which arranged according to the (j's, give as determinant of the coefficients

a. -as a. -al

as a. - al - a, - a2 al a. - as

a] a2 aa a. , the (+ l) ((-transformation and arranged according to the a's

- fJ. fJa - {j2 {jl

- fJa - fJ4 fJI fJ2

/'j2 - fJ1 - {j4 fJa /11 fJ, {/s fJ. ,

the (- l) {j-transformation, We can now &tate the following: If the equiangulaL' double rotation to the right (.7t'1 3(. 3(3 3(4) transfers

the 'Vector (UI a2 a3 a.) into (fJl fJ2 ~8 I~.) then the (+ 1') a-transformation transfers tbe vector (.7t'I .7t'1 .7t'a .7t'.) into ({jl fJ2 (J3 fJ.)

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and the (- r) iHransformation transfers the vector (~1 ~, ~J ~4) into (al a~ a 3 ( 4 ).

Analogous to this: If the equiangular double rotation to the left «(>1 Q, (>8 (>4) transfers

the vector (al a2 a3 ( 4) into (~l (j, (j3 ~ 4)' then the (+ l) a-transformation transfers the vector «(>1 (>, 1?3 (4) into «(jl (j, (j3 (j4)

and the (-l) (j-transformation transfers the vector (Ql Q, Q3 (>4) into (al a, a3 ( 4).

Let us now suppose that S4 has first an equiangular double rotation to the right (n) transferring an arbitrary vector a into {j'; then an equiangular double rotation to the left (Q), transferring (j' into "I, then we can write:

n = [(+ 1') a]~' v

3t = ([(+ 1') a]

Q=[(-l)"I]~'

[(-l)r]11?

where the form between I1 denotes the determinant of transformation having as first row the sum of the products of tbe terms of the first row of [( + 1') a] with tbe corresponding ones of respectively the first, second, tbird and fourth of [(-l) y], whilst the second etc. row in a corresponding manner is deduced out of the seoond etc. row of [(+1') al (all this in the way offorming a product of determinants).

If S4 has first an equiangular double rotation to the left CQ) transferl'ing a into (j" and then an equiangular double rotation to the right (~'),

transferring ~" into y, we have:

Q=[(+l)a]~" 3t'=[(-'l')y]~"

n'=([(-'1')y] . [(+l)a]IQ. But now

[(+1')a]. [(-l)y]=[(-1')"I]' [(+l)a]

which can be at on ce verified, so:

~'=3t.

Thus: If S4 is allowed successively an equiangular double rotation to the

right (.1l) and one to the left (Q) the order of the two rotations may be interchanged. For, in both cases an initial position of a vector a gives the same final POSitiOll y.

A d 'f d th d I {J' R" tb (a (J') . n 1 we regar e qua rup e a, 'IJ, "I, en {J" "I lS

equiangular to the right, according to the rotation (~) and (;, ~") equiangular to the left according to the rotation (Q); by which we assuredly once more have proved theorem 1.

48 Proceedings Royal Acad. Amsterdam. Vol. VI.

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Let us farthermore suppose (IJ) and (T) to be two equiangular double rotations to the right, transferring a given vector E the for­mer into ;, the latter into 11. Then

a=[(+'1')E]; ~=[(+'1')E]11.

The same orthogonal transfol'mation transferring a into ;, transfers Tinto '1'], so that the angle between ~ and 'I'] is the angle bet ween the vectors a and T independent of the initia] position E. As a special case theorem 3 is included in this, for the case that the two double rotations take place about an angle t:r; fol' then the angle between ; and 'I'] is the angle of the two planes of l'otation through E, proving to be independent of E.

We have still to mention that theorem 1 in the second form is entirely included in the applications of the blquaternions on S4 as given by Dr. W. A. WIJTHOFF in his disseI,tation: "De Biquaternion als bewerking in de ruimte van viel afmeting.en" (the biquaternion as an operatlOn in fourdimensional space). For an equiangular double rotation to the right is represented by Q. El + E, (p. 127) where Q l'epresents a certain quaternion with norm unity.

This applied to an arbitrary double v('ctor

alEl + a2E"

changes 11 into QalEI + a2E"

SO it leaves the isosceles part of that double vector to the left unchanged and so also the equiangular system of planes to the left to which it belongs. This holds good for an arbitrary double vector, so particularly for aplane.

Fmally theorem 3 can be proved as follows: If tp and 1J' are the acute angles of position of two planes, then

if we represent the coefficients of position respectively by .ts and f-L's:

cos tp cos", = 1.28 f-L2B+I.S1 f-L81 +Î.12 f-Lu+Î.14 f-Lu+ l ,. t'u+lu f-Lu = = ~ P'n+ÀI4) (f-Ln+f-L14) - ~ (lu f-L14+J.u f-Lu)'

Far intersecting plan es with angle of positian tp:

C08 tp = ~ ().23+l 14) (f-Lu+f-L14) = ~ (lu-lu) (f-L2S-t'14)'

80 for two intersecting planes, belonging to two definite equiangular systems to the right Ol' two to the left

l'sa+À'u=)."n+l"u and f-L'n+f-L'u=f-L"U+f-L"14

" ~, -'" ,,, d' , - " " d ' " resp. '" u -'" 14-A U-A 14 au f-L u-f-L 14-f-L 28 -t' 14 au cos tp = cos tp •

We shaU now resume our geometrical reasoning dropped aftel' theorern 3. Let us take through 0 a definite vector 0 W in S4 but not movable with S4 and let us represent each system of planes equi-

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angular to the right by the line of intersection of the planè through OW belonging to it wIth Sa 1. 0 W. That Ss is thus, entirely fiUed with these representing lines which are in (1,1)-correspondence with the represented systems of planes~

We shall call that Sa 1. 0 W regarded as a complex of the rays repl'esenting the equiangular to the right systems of planes, "the repre­senting Ss to the right of S/' or shorter "the Sr of S4". In the same way we form the "St of S4". Earh pair of planes in S4 is then unequivocally determined by its representants in Sr and Si and rever­sely the pair of planes determines unequivocally its representants.

Theorern 4. An equiangular double rotation to the right of the S4 ab out a certain equiangular system to the right which double rotation leaves according to theorem 1 the position of St unchanged, gives a rotation of Sr about the representant of the system of the planes of rotation over an angle equal to double the angle, over WhlCh the equiangulal' double rotation to the right of S4 takes place.

'P' p' Proof. In the first place ensues from theorem 3 that the representants in S, keep makmg mutu­ally the same angles; so Sr has a "motion as a solid". We suppose through OW to be laid its plane of rotation a in S4 and its norm al plane fJ. In fig. 8 we suppose the lines tendmg downward to lie in a and those tending upward to he in fJ

e wIuIst the indicated directions of rotation are those of the equiangular double rotation to the l'ight which we consider. Angle WOeis made equal

Fig. 8. to 1 ~. Then the Sr is the Ss through oe and (l Let OP be an arbitrary vector in fJ and lP the angle, over WhlCh the equiangular double rotation to the rlght takes plare. The double rotation leaves oe unchanged as representant of the equiangular system to the right on (afJ). Moreover it transfers OWinto OW' and OP mto OP'. If we then still make L pil OP' equal to LP' OP we have:

(OP, OW,

Opr

) ow' equiangular to the right, thus:

(OPr

,

OW, OP ) OW' equiangular to the Ieft, Ol' also:

(OP", OW,

OP' ) OW' equiangular to the left, so at last

(OP", OP',

OW) OW' equiangular to the right.

48*

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The plane POW glvmg OP as representant of its equiangular system to the right before its double l'otation, gives aftel' that rotation (transferred to P' 0 W') as representant OP" making an angle 29' with OP; so an arbitrary vector OP in SrJ.Oe (the invariabie vector) rotates about oe over an angle 29', with which the theorem is proved.

We can now say: For an S4 moving as asolid about a fixed point the position is at every moment determined by its "position to the right" (the position of the Sr moving as asolid about a fixed point) and its "posltion to the left" (the position of SI). For, if of two positions the pairs of planes through 0 coincide, tben this is the case too fur all planes, thus for all rays too.

N.B. We can remark by the way, that in this way we have proved quite synthetically that two positions of S4 have a common pair of planes, namely that pair, which has as representants the axis of rotation of the two positions to tbe right and that of the two positions to the left; so, taking into consideration that also the common fixed point is always there (having a& projections on the positionó of planes remained invariable the centres of rotation of the projections of S4 on it), that the double rotation is the most general displacement of S4. However, until now we have occupied ourselves onIy and wisb to keep doing so with the motions of S4' where always tbe same point 0 is in rest.

For a continuous motion of S4 the position and the condition of motion are at every moment determined by Sr and SI; so the motion of S4 is quite determined by the motions of Sr and SI; and at every moment the resulting displacement of S4 is quite determined by that of Sr and of SI, independent of the order of succession or combining of the two latter ; they have no influence upon each other. We can regard a motion of S4 as a sum of two entirely heterogeneous things, i. e. which cannot influence each other in any way or be transformed into each other.

We can proceed anothel' step by indicating not onIy by a line in Sr a system of equiangular planes of rotation to the right, but also by a !ine vector along it an equiangular velocity of rofation to the rigbt, that line vector being equal in size to the double velocity of l'otation of the double rotation and directed along the vector moving with S4 in the direction of 0 W. Then equal and opposite velocities of rotations to the right of S4 are indicated by I1qual and opposite vectors in Sr.

Let OPr be snch an indicating vector and OQr and OSr two Ulutnally perpendicular vectors in the plane erected perpendicularly

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to OPr in Sr in such a way that

(OPr. OW) OQ" aSr

is equiangular to the right, then to the equiangular double rotation to the right of S4' indicated by OP, correspond~ the rotation of S, about OPr equal to the length of OPr in the direction of OQr ~ OSr.

Let OPr be another indicating vector and let us determine in an analogous way (JQ', and OS'n then the orthogonal systems ofvectors

o WPr Qr Sr and

OWP' Q' S' r r r

are congruent, can thus be made to coineide by a rotation of S4' with 0 W, thus Sr too, remaining in its place, so that the indicating vector Op,. by a motion of S, in itself ean be made to coincide with the indicating vector OPr in sueh a way, that at the same time the directions of rotation of Sr belonging to it coincide in the normal planes. But then an indieating vector in Sr represents that velocity of rotation entirely in the way usual in spaee of three dimensions, as also by its leng th it indieates the size of the velocity of rotation of Sr belonging to it; if namely we endeavour to regard the definition of that usual mode of representation entirely apart fJ'om the notion "motion with Ol' against the hands" whirh is lacking in 84 ; and say simply aftel' ha ving taken in that space a system of eoordinates OXYZ: the vector of rotation will be erected to th at side of the plane of rotation, that for the plane of rotation being by motion inside the space made to coincide with the plane of XY in such a way, that the direction of rotation runs from OX to 0 Y, the vector of rotation is dil'ected along the po.3itive axis of Z

For us that system of cool'dinates in Sr: OXr, OY" 02,. has been chosen in such a. way that with 0 TV it forms a system of coordinates in S4' for which

(OY" OZ,) (OZr, OXr) (OXT! OYr) OXr, OW' OYn OW' OZT! OW

are equiangular to the right. A vector along OXr then indicates a rotation of Sr in the sense

of OY;. ~ Oz,. Entil'ely analogous reason\ings hold good for S,. It being however

lnore profitable to be able to say also for SI: a vector along 0 XI repl'esents a velo city of rotation of St in the sense of 0 Yt ~ OZL, we must modi(y the preceding either by choosing the system OXI fz Zt W in sllch a way that

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(OY/. OZt) OXi. OW

is no more equiangular to the right, but to the left, or if we suppose

(OY/. OZt) OXI. OW

to be also equiangular to the right, we must take as indicating vector in Si that vector in the direction of WhlCh 0 W would move together with S4' not the one moving together with S4 in the direetlOn of 0 W. We shall do the latter. The advantage of this choice will be evident from what follows.

We have stIll to remark, that if only the position of Sr and St is determined, the position of 84 ensues from it not in one, but in two ways; for, a position of S4 gives no other positions of S, and St as its "opposite position" for whieh all vectors are reversed; that opposite position can be obtained by an arbltrary equiangular double rotation over an angle 3'(; Sr and Si then rotate 2 3'( and are again in their former positlOn.

But a continuous motion of S4 out of a given initial position is unequivocally determined by the given continuous movements of Sr and St out of tho corresponding initial positions. 80 we shall have completely answered the question how asolid S4 moves under the action of determmed forces if we ean point out how Sr and Si move undel' those actions ; in other words if we ean point out "the cones of rotation in the solid and in space" .

APPLICATION., The Euler motion in S4'

The equations of motion for this have been given for the first time by FRABM in the "Mathematische Annalen" Band 8,1874 p. 35. The system of coordinates OX1 X 2 X a X 4 in the soUd we shaH choose in such a way th at the produets of inertia disappear. We shaH suppose

(OX2,OXa)

OX1, OX4

to be equiangulal' to the right. And we choose OXr y, Z, and OXi y, Zi in such a way that:

( OX1' OX2 , OXa, OX4 )

OX" OYn OZ" OW

is equiangular to the l'ight and

( OX1' OX2 , OXa, OX4 )

OXl, OYl, OZI, OW

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( 729 )

equiangular to the left, (from which ensues as a matter of fact that also

(OXn OYr) and (OXI, OYt) OZn OW OZI, OW

are eqtlÎangular to the right). The systems OXr Yr Zr and OXI Yz Zt execute the motions of

Sr and St which are to be considered. Let us farthel' eaU 2(1)3' ,(1)1' 1(02' 1(1)4' S(l)4 'S(l)4 the components of

the yelocihes of rotation according to the system OX1 X s X s X 4 ;

and CP1' CPs' CPs the components of 'the veloclties of l'otation of Sr according to OJ{" 0 y" aZr, hkewise tfJ1' tfJs' tfJs the components of the velocities of rotation of SI accordmg to OX1, 0 Yl, OZt. Then we know the components of velocity of rotation to the right

OXs ~ OXa • OYr~ OZr 1- (,(1)8 + 1(1)4) according to OX

1 ~ OX

4

01' accordmg to OXr~OW

and analogues, and hkewise the components of velocity of rotation OXs-:' OXa to the left 1 (,(08 - 1(04) accordmg to OX

4

-:, OX1

or according to

OYI~ OZt o W ~ OXI and analàgues.

Therefore:

CP1 = 2(1)a + 1(04

cp, = a(Ol + S(04

CP1 = 1(02 + 8(04

tfJl = ,(Oa - 1(04

tfJ, = a(Ol - ,004

tfJa = 1(0, - aWf'

The Euler equations of motion "in the solid" (giving the opposite of the apparent motion of the surrounding space) follow more simply than accordmg to FRAHM out of the vector formula

fluxion of moment of motion = moment of force - rota­tion X moment of motion

which is easy to understand for a three dimensional space as well as for a four dimensional one,

(and where the sign X indicates the vector product)

Fol', "in tlle space" the fluxion of tlle moment of motion = moment of force; but of this for the position in solid has already been marked the fluxion wanted to keep the position constant in the solid, 1. e. the fluxIon w hieh corresponds to the rotation of the moment­vector about the rotationvector and this is = rotation X moment of motion.

Let us eaU the squares of inertia :'!: m ,'111' etc. PI etc. and let us put

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------------- ---

( 730 )

PI + P~ + Pw + P 4 = R

- PI + p~ + Pz - P4 = Al PI - P~ + Ps - P4 = A~ PI + P~ - Ps' - P4 = As'

Then we ean write the rotationvector in the form:

~W3 i + aWlj + lW~ k + Tt (lW4 i + ~W4j + BW4 k) or in the form

El (PI i + CfJ2 j + (Ps k) + E~ (t/Jl i + t/J, j + "'8 k). The notations hand E are faken from the above-named dissertation

of Dr. W. A. WrJTHOFF; h is defined on page 67; El and E~ on page 78. The moment of motion heeomes

(P~ + Ps) ~ws i + (Ps + PI) sWI j + (PI + P,) 1 W, k + + h (Pl + P4) lW 4 i + (P2 + P4) ,w4 j + (Ps + P4) sW4 kl

or in an ofher form

! El (Rtpl + AI t/JJ i + (Rep, + A, '1',)j + (Rrr 3 + A3 'I's) kj + + 1 E~ (RtfJl + Al (( 1) i + (Rt/J~ + A, p~) j + (Rt/Js + As 'r s) kj.

If (p and t/J represent the rotation veetors in Rr and Rl, we ean write the rotation:

and the moment

1 R (El (P + E, t/J) + U El' (A) t/J + 1 E,. (A) cp),

wh ere the notation (A)rp means: Al (PI i + A~ rr,j + As Ps k. The fil'st and strongest of these tel'ms falls along the rotationveetol';

for a body with equal squares of inertia it is the only one; the seeond, whieh, together with the A's, becomes stronger as the body is more asymmetrie, we might eaU the "crossed moment" fbecause its right part is caused by the left part of the rotation and inversely.

Let us put finally the moment of force in the form El I-" + E~ v, where I-" and vare threedimensional veetors ; then the above given formula of the vector ean be brok en up info the six C following components, given suceessively by the coefficients of El i, E~ i, El j, Es j , El k, f, k.

R~l + Al tÎJl = A~ t/J, Pa - As tf.'s' p, + ,21-"1

R,pl + Al PI = As cp, t/Ja - As Ps t/J, + 2vl

R~, + A, tÎJ2 = A3 t/Js PI - Al t/Jl (ja + 21-",

R;P, + A, p, = Aa CPa '1'1 - Al PI "'a + 2v2

R~a + Aa tÎJs = Al t/Jl P2 - A, t/J, 9\ + 21-"a

R;", + Aa Pa = Al CPI 'l/', - As cp, t/JI + 2v"

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( 731 )

If we put R' - AP' = ap and RAp + Ag A,. = bp, and if we represent the vector al CPl i + a, cp, j + al CPa k, by (a) cP, we can write the six equations of motion :

(a) ~ = V. (b) tp • cP + 2Rf.t - 2Av

(a) tP = V. (b) (jl • tp + 2Rv - 2Af.t.

For absence of extern al fOl'ces:

(a) ~ = V. (b) lJ' . cp j (a) tP = V. (b) cp • tp

In this form we can directly read:

. (h)

1st• lf in the initial position cp = lJ', then cp remains equal to lfJ, i. e. if the initial l'otation of S4 is a rotation / / to a principal space of inertia, then the motion remains a rotatlOn / / to that space. The equations of motion for that case can be reduced to a system to be treated as the weU known Euler motion in Ss when the forces are missing.

2nd • For unequal A's "invariable rotating" is only possible under the following two conditions which are each in itself sufficient:

a. for cp and tp both directed along one and the same axis of coordinate (X-, Y- or Z-axis of the representing spaces) i. e. fol' a double rotation- of S4 about a pair of principal p[anes of inertia i

b. tor cp = 0 or tp = 0, i. e. for an equiangular double rotation of S4'

It has been pointed out by KÖTTER (see "Berliner Berichte" 1891, p. 47), how a system of equations analogous with what was given, can be integl'ated. (The pl'oblem treated there is the motion of a solid in a liquid). Accorrling to the method followed by him the six components of rotation can be expressed explicitly by byper­elliptic functions in the time. If however we have CPl' Cf" !fs' tpl' tp" tpz expressed in the time, we have the "cones in the solid" for Sr and St. To deduce from these the "cones in space" we set about as follows. We notice that the moment of motion to the right Rf/' + (A) ti' in St remains invariable "in space" (in SI that vector changes of course its direction, but there R~, + (A) p remains invariabie) ; calling the two spherie coordinates ("polar distance" and "length") of p with respect to Rep + (A) tp during the motion of lp in space {)o and X and l'emal'king that each element of the "cone in space" at the moment of contact coincides with the corresponding element of the "cone in

the solid" J we ean express {)o, .iJ. and i in the time1 with which the

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( 732 )

tlcone in space" for Sr is found. Analogously the "cone in spaee" is determined for SI with respect to Rtf" + (A)p.

We shall just show that as soon as two of the squares of inertia P become equal, whieh means the same as two A's beeomillg equal we can do with usual elliptie functions only.

For instanee let A, = Aa, thus also a, = aa = ,as; b, = ba = ,ba. Let us eaIl ,lP~ and ,'/1a the value of the decomponents of cp and lJJ in the YZ-plane, Frp and Fof the anomalies ofthose deeomponents (counted from Y to Z), and F the diiferènce of anomaly of ,l11a and 2 ra; then the equaHons (h) become for this case:

Pl , ,lP8 , tlJl , ,'1>8 and F, namely Cl

Pl + tlJl = -V . . . . . al

. . (1)

,a8 ,Pa' + al Pl' = c,' .. (I1) , ,aa ,'1>8 s + al '1>1' = Cs' ,aa ,ba ,Pa ,1/'a C08 F + al bl Pl tlJl = C4 • • • •

(III)

(IV)

If we put in these Cs

,Pa = -V C08 11, ,aa

ca ,'1>a = -- cos~,

V,aa we have the diiferential eqations

,ba ,. . 1] = - V ca cos ~ s~n F,

,aa al with the two integrals:

c, sin 1] + Ca sin; = Cl'

,ba C08 1] cos ~ C08 F + bl sin 1] sin ~ = ~, . C, ca

so that af ter elimination we obtain the following differential equation in 1]:

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( 733 )

or c, sin ?j (= V al • PI) put equal to u:

,aa' al ~, = F4 (u), in which we can easily verify that F 4 (u) has two real roots between - c, and + c, (the two other roots are rea]

outside those limits Ol' imaginary according to ,b/-bl ' being ~ than 0);

those two roots indicate the limits between which fj swings to and fro according to a course indicated by elliptic functions. For the case ,bB' > bi' fol' in stance, thus for tour real roots UI < U, < Ua < u4 •

that course becomes:

and

Fal-thermore:

where the second member is a rational funètion of lfl (lPl can be rationally expressed in PI according to (I), ,Pa' according to tIl), ,Pa ,'l/'a cos F according to (IV»), so that F'f too can be expressed in t by elliptic functions and by that the entire "cone in the solid" ; and further according to the above method also "the cone in space".

The following special cases can very easily be traced to the end. 1·t • The four squares of inertia are equal two by two. This case

is obtained by putting As = Aa = O. Then

al =R' -Al'

,aa = R' bI = RAl

,ba = O. • And the equations of motion pass into:

al PI = 0 ,aa ,Pa = 0

. bi F'f = -lPl

,aa

al tPl = 0

,aa ,..pa = 0

. bi Fo[J=-PI

,aa

• bi F = - (PI - lPI)

,aa

from which we directly read, Pn ,fj!" 'l/'l and ,"'a remaining constant, that the "cone in the solid" for Sr and fol' SI is a cone of revolution with the X axis as axis of revolution. FarthermQre the moment

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x

z Fig. 9.

( '734 )

Rq:>+Altpl to the right lying in the meridian plane of q:> l'emains in Sr invariable. Thus "in space" that meridian plane rotates about the vector Rq:> + Al '~l> by which the "cone in space" is known, and likewise proves to be a cone of revolution. Analogous for St. Fig. 9 shows the two con es of rotation

The outer cone is tbe moving one.

2nd• Three of tbe squares of inertia are equal and unequal to the fourth. We take the axis of the unequal one as x.1 axis in S4' Then Al=A2=Aa=A; al =a2=a3 =a; bl =b2 =ba =b; anel t11e equations (h) pass into

. ó cp=- Vtp.q:>

a

. b tp=- V!p.l~

a

therefore;' and tP ale both perpendicular to q; and to 11', wbilst rpttP=O, so cp + tp is constant and <jJ and 1/' are each for itself constant in absolute value, so th at tbey both rotate about tbeir sum ("in space" th at vector of the sum has in general quite a different position fol' Sr than for St) by which the two "con es in the solid" are determined. "InvariabIe rotating" of S4 we have here wberever lp and tI', ,-

o

l!'ig. 10.

regal'dless of their yalue, C'oincide. To find the "cone in space" for s" we notice the inval'iability in Sr 0 of Rq:> + Atp. "In space" ,p rotates about R'p + Atp, fol' the angle

~I between those two vectors remains constant. In St rotates analogously "in space" ti' about Rtp + Alf. Fig. 10 represents the two cones of rotations in Sr. (Here too the outer cone is the moving one).

We remind the readers once more, th at w here we bring <jJ and 1/', as far as their positions in the solid are concerned, into relationship with each other, we must of course in our mind make the positions to the right and the left that is of tlle systems of coordinates OXr

1';. Zr and OXt Yt Zt to coincide with each other, so that but one system of coordinates OXYZ is left (for instance for t11e equations

- 21 -

( 735 )

(/t) this must be noticed); but in space (i. e. the Ss J. 0 W) those systems of coordinates OXr Yr z,. and OX/ Y/ Z, have at earh ,moment very definite positions differing from one another.

Chernistry. - Prof. C. A. LOBRY DE BRUYN read aIso in the name of Mr. L. K. W OLFF a paper entitIed: "C'an the presence of t!te molecules in solutions be pl'ovecl by application of the optical met!tod of TYNDALL?"

(This paper wiJl not he published in these Proceedings).

Chernistry. - Prof. C. A. LOBll.Y DE BRUYN presents also in the name of Prof. A. F. HOLLEMAN a paper by Dr. J. J. Br.ANKsMA, entitled: "On the sltbstitution of the core of Benzene."

(This paper will not be published in the Proceedings).

(April 19, 1904).