part
new
to
witli
that
much
stress
on
the
without some
is
hardly
same
time,
expedient
to
make
frequent
use
of
alternative
&
Harmonic mean
more than
two roots
sign
q
things
are
the case
theorem
141
of all
and negative
298
The
penultimate
convergents
of
.
—
—
-
—
-
...
5.
measure
their
ratio.
Thus
exactly
measure
the
required
13.
If
-
solved
by
the
:
the
ratio
5
: 37
to
make
to,
or
less
than
the
—
•
work
done
to
9,
how
times
the
coffee
and
16
of
tin.
A
fused
9
minutes
b ,
Inen,
by
deimition,
are known,
4
hours,
2
of
Venus'
revolution,
assuming
the
distances
of
the Sun to
C,
when
A
per
cent.
3
case
is
8.
one of
which varies
distance
from
the
starting
point
of races
these, and
how long
of a diamond of one carat is
the
has served 9
How
long
had
they
served
pensions
is
343
2s
have
three
useful
formula;
(1),
(2),
(3)
of the letters
to
the
of
a
and the
terms will
;
12,
th
term
is
4?i4-
1.
24.
we have
8
terms
P. are
number
of
number of
is even
product
of
the
first
set
same
for
p
as
for
q
terms,
shew
series
terms,
remember
both
forms
given
above
for
s.
using
(2)
series
is
always
less
than
2.
series
respectively,
prove
as
follows.
Tojind
the
value
of
a
recurring
decimal.
Let
integral
number
consisting
of
the
nonrecurring
terms
which
involve
r
n
can
be
integer, shew that
LAlt
- bb
proved it will
shew
that
a
: a
be a, b,
the
(n—l)
—
in
1.
A
breadth
length
top layer of
1089
?
a complete rectangular
rectangular
pile,
11 and
course being
half the number of shot in
a square pile,
courses when
14.
If
the number of
number
r,
of
which
any
powers
of
:
5
and 8127
method.
Example
3.
In
what
scale
will
the
nonary
number
number
EXAMPLES.
VII.
b.
1.
eleven.
9.
Transform
number
182
denoted
by
222?
25
21.
In
8 if the
where
q,
that
values of
ci
Note.
this assumption,
that
21.
G
24. 21+3
a
4
b
J
square
b
s
we must take
retain
the
symbol
J
110.
We
2-5\/
r
l
2
therefore
ft
and
c
t
form
the
equation
whose
roots
—
of
the
corresponding
quadratic
equation
y
2
long
as
a
in
theorems and examples.
x, is
two
variables
x
and
y
ax
2
the
equation
ax
2
ratios
and
equate
it
to
the
product
of
the
ax
2
quadratic
will
satisfy
the
equation
ax
2
+ bx +
c
x
be
solved
in
J(x
used when the
equation
(3)
equation
is
We
(5).
Hence
from
(4)
and
(5),
by
division
y-*-
3
1
later,
the
number
of
solutions
is
limited.
If
the
number
of
unknown
quantities
p
of
the
equation,
and
by
x
and
y
are
admissible
the
there
of
each?
Let
x,
y,
z
equation
is
£5
paid
in
half-crowns
only?
19.
Divide
136
which
the
two
operations
making
the
two
journeys
operations
;
2,
or
24.
141.
To
find
n
P
r
a
railway
required
answer
formed by
Note.
The
required
number
by
forming
all
the
groups
containing
2
Americans
and
4
Englishmen
the
number
1 ?
How
drafted
12.
only occupy
10 persons ?
of 3 Latin and
men volunteer,
in how
many ways
of
10
books
be
things
have
been
regarded
regarded as
;
and the rest
to be unlike.
x
permutations
we
should
obtain
x
x
\p
permutations.
repeated
once,
twice,
up
to
r
ways in
which the
a
prize.
Thus
two
prizes
can
the
following
manner.
time is greatest.
all
possible
ways
each
of
the
foregoing
groups.
(1)
gives
and
decorated
with
fourteen
flags
the
letters
relative
positions
of
a
railway
station,
animals,
and
to
be
shipped;
in
how
many
ways
possible
three
distributed,
once
I
15.
10000
of
money
from
a
bag
a
penny,
them
may
together
same straight
using the
required
to
prove
that
the
;
a,
b,
c,...k;
p.
2
+p
1
at a
-
the
we
the same
n
factors.
The
highest
(x +
a)
+
two
the
corresponding
terms
of
the
first
expansion.
Hence
the
value
is
of
n
dimen-
sions,
being
a
product
formed
by
multiplying
together
n
letters,
one
taken
of
combinations
expand
expressions
the
series
(2),
we
proceed
thus
c
x
x
integer;
be
for
all
values
of
m
and
n.
The
principle
of
in any
not
appear
terms
when
tive
factors,
we
may
write
of
(1
sum of
first
r
terms
of
the
1
x
r
series
(1)
the
series
y.
given
expression
correct
to
5
places
of
be greater than
;
things
r
at
a
time
when
repetitions
combinations
r
in
the
a
positive
integer.
11.
Shew
that
that
I//-1
a)
c
-c
l+
c
2
-c
3+
+(-mv-(-i)'-
1/
2>=/3
20.
which
suppose
a:
12.
great
advantages
characteristics
can
be
(2)
The
mantissse
numbers
so
This proposition we
proceed to prove.
dividing
number
is
KM) *a-K
'8821259
; write
the
integral
part
of
the
numbers
transformed
to
base
10.
In
base
to
which
Napierian
first
-, 1 1
of
y'
2
in
216]
of
the
1
common
of decimals.
practice
227.
The
following
form
4m.
21.
instead of the
we
have
R
amount
renewed
every
half-year
otherwise
stated
we
number
of
years
ever
it
is
called
a
perpetuity
At
the
end
Value
of
of M,
equal
to
which
interest
is
reckoned
a
£20000; what
of
the
annuity;
then
since
.R
17
th
year;
having
given
log
2
are
equal.
4.
action.
We
have
a
1
and
1.
Consider
the
ih
powers
decreases without
denote
the words
of
Mathematics
precisely their use
or
unlimited.
It
will
therefore
be
convenient
a
lf
a
si
o
3
+ S
fortiori
the
pro-
position
taking x small enough ive may make any term as
large
as
we
please.
of
terms
in
descending
powers
of
x,
by
taking
x
small
it.
Example
1.
=-=
:
—
if
the
coefficient
ofx
2
Hence
the
given
series
is
convergent,
than
some
finite
quantity
a,
the
sum
of
student
should
notice
the
all
the
terms
are
of
the
same
sign
is
diverge)it
if
from
and
after
some
fixed
term
n
series
-
negative
when
x
is
same
sign;
and
therefore
a
fortiori
it
is
convergent
when
examples
are
important,
and
will
be
required
—
that this series is
(l—x)~
introduce
the
infinite
to be
are
(1)
.
&
the
?i
th
term
to
the
'preceding
u v
tively; therefore if
first.
term is
n
240
1
divergent.
n
*307.
We
have
terms
in
A
and
B
are botli
A
n
B„
as
in
the
former
case.
Let
A
that
by each
311.
If
two
)
investigation
q
o
values
of
n,
the
coefficients
to
assume
2. 1.2.
factor of
x,
y,
z
when
tit
en
each,
coefficient
must
be
are
equal.
Suppose
that
the
?/
jwtial,
of x
u
 
we must
second
we
must
have
one
more
term
ABC
h
1—
curring
series
a
integers
the numerical
p
n
q
nl
1. 2
the
suc-
cessive
convergents.
17.
approximates
to
of
two
quantities
4
un-
knowns
x
and
y
can
be
reduced
to
be
limited.
It
a,
en
cd
numerically
smaller
two quantities -j-
is
the least value
t can have
by
order
the divisions which
and
third
bells
tolled
; how
integers.
22.
Find
the
the
1
the
sixth
convergent
to
each
1.
v
/3.
2.
^5.
3.
y/6.
4.
s/8.
5.
v/11.
6.
x
/13.
7.
x/14.
8.
V22.
—
;
-
thus
a
n
cannot
be
greater
than
a
v
359,
a
recurrence
must
take
place,
and find the
1
J_
J_
J_
13.
U
X
number of
w
r
e
please
82 127
3, 2,
we obtain the solutions
discover
for
what
values
of
variables
are such that
if their product
m
and
n
are
either
both
adding
their
21.
buy
hogs
hog; Hendriek
bought 23
11 more
x
of
Art.
-
by
the
follow-
ing
rule
the numbers
numbers
Pascal
with
inclusive.
'a
1
For
numerator
in
ascending
order,
n
terms
infinity
to
the
(n
of
the
third,
fourth,
fifth,...
orders
of
differences,
the
left.
Thus
u.
2
obtain
Au
3
the
forming
the
orders
of
differences
we
eventually
come
to
the
terms
are
equal.
Let
order
of
number
of
geometrical
progressions
not found
while to
of
differences
A and
G.
9
Example
4.
Find
the
sum
of
a
to extend
,
article is most
(n
:
number
in
a/3y<$...
are
equal
to
required
the
of
ways
is
-(1
a
third
time,
namely
theorem
investigated
and
the
last
shew
that
3,
except
7,
shew
that
n
G
431. To
and
once
negatively
among
the
(^;
—
true
universally.
a
given
number
N
and
than JV and
greater
it
may
be
shewn
that
2
in
ascending
order
of
magnitude
*443.
fraction
a,
/3
the
by
ka,
where
k
is
an
6
x
6,
—
.
1
ways
JL
events
A,
B,
C,...,
of
which
one
must,
and
only
Probability requires
Probability,
a
queen.
3.
the
one
is
two-thirds
that
odds
in
favour
of
events
and the
article
by
supposing
P
1
=P
2
=P
3
13
C
3
and tail
13
queen
is
12
then
throwing
9
is
throwing 10 is
the
drawing
a
is
been:
find
the
chance
that
the
here
shillings and
the
chance
(1)
that
a
card
is
of
each
suit,
(2)
that
19.
A
to
be
ridden
by
chance is
white;
a
ball
is
drawn
and
replaced,
another
is
then
coins removed
=
throw with one
A
can
have
a
infinite
series
5MKi©
4+
pick
out
the
coefficient
of
x
p
and
was
published
by
him
in
1730
j
2.
A
are
a
regular
tetrahedron,
the
case of the tetrahedron;
of
throws
is
continually
increased
the
that each of
former are
usually called
of the
existence of
the r
from
another drawing
make the
the
statement
made
by
A
and
B
the
probability
that
it
is
false
as
pp'
to
c
(1
—p)
(1
that
the
quantity
c
becomes
containing
6
balls
all
of
different
colours
(ii)
one of
A
two heads
:
the
three
numbers
prize,
while
C
,£1
prize
B, or by
1,
all
black ball
specified coins.
be
n
things
named
until
winning,
:
an ace
wants
looks
win if he
or
5
or
4,
and
so
on
dummy
to
throw
in
the
prove
that
their
chances
are
respectively
/8\
2
56 .
/7\
2
(ja)'
W>
and
out
of
divided into three parts by two
points
within
the
larger
shall
not
exceed
a or b.
and in
be
found
that
determinants
may
often
see
that
a
i
into which this
may be,
and so
symbols in order that
the
original
definitions.
At
the
same
in
such
proved
that
the indices
and
with
perfect
generality.
512.
Equations.
514.
To
find
the
by
writing
down
only
the
coefficients
being
are
rational
and
integral.
also that the product of
a
contains (b
3
-2^
2
z*'
xy(x
the form
A (x
a
3
the
equation
plicable to
and
to
Chap.
xm.
of
tion
take
is
due
to
Euler.
it
follows
that
we
can
eliminate
n
and adding,
equa-
satisfy the
a
2
the squares and
 
negative
roots
of
f(x)
sign in
7
of
roots
will
of
a.
Let
see that if
 
the
reciprocals
of
the
roots
of
the
proposed
equation.
Let
y*
degree.
Hence
any
Putting
let
11
1
a
whose roots
are the
the original
the roots
of x
Solving
this
the
equation
x
3
are
y
and
w
2
the
other
x*
:
investigation
fails
=
\
J6
p
3
two
the
such
that
stream.
17.
Extract
of
arrived l£
hrs. sooner
: find the
length of
into
a
in
each
: required the number
operation.
55.
and
if
3,
1.2.
M
CV2.3.'
l
C_
1
£3
+
has
its
terms
of
an
arithmetical
took
a
+
the squares of three
are
123.
If
«x,
a
2
y
a
3>
a
i
are
an
y
of X.
(2)
1,
unequal,
b
cannot
Woolwich.]
146.
of a
year was
at
the
end
whole was
of
cotton,
cwt.,
£508.
7560 numbers
is 10560.
and
6.45,
and
same pace
as A,
C
town
at
: shew that
sum of
one
less
than
half
the
company
; while
the
series
score
to be white;
straight
line
in
which
a
candidate
may
obtain
rent
r
ave
taken
under
of the third
36f.
If
7
by
by
each
killed
by
each.
If
A
had
fired
as
often
as
B
and
B
as
20. 3,4,1.
greater than 1. Tho
+
11
21
12.
6. 40320;
13.
flic
at
least
four
MISCELLANEOUS
EXAMPLES.
Pages
(2)
^/-g-
is
London.
T
*