of the same class of readers,
I have
volume
finds
an
apology
in
public
engagements
that
as
Integral
Calculus
recollect
common sense.
has
been
useful
Every
mathematical
book
that
When
of
the
work
will
worked,
remembering
the
45
=
Numbers
97-109
The
Rational
OnefSld
99
Dedekind's
PRODUCTS.
•
• •.
'^^^
Algebra to
148-157
Condition
for
Series
149
Circle
and
Radius
of
Convergence
149
Cauchy's
Introduction
of
u, sech
Integers
Terminating, Non-Terminating,
Becurring or
H<sjC
. .
Theorem that
a Prime
Integral
Squares
Every
Partial
Criterion
r
out
of
n
occasions
More
a
of
be,
ac,
ab.
C.
II.
1
to is that in
that
once
manner of
chap, iv.,
the
student
may
fill the
first
letter
ai
^
n~r+iP*i
in
We
that is
is
now
no
restriction
on the choice of the letter. Hence the first two may
be
we may
combine any
the
will thus
only
only
on
the
arrange his boat
of 4 things, that
Since the arrangements of
ways of
4
The
each
would
got
all the
enumeration
The theorem (when
we
*
 ^
or
1 times.
This new
problem may
it
the
letter
s
(initial
of
same)
as
often
nations the
can arrange
in
which
a^
occurs
at
least
once,
the
number
the
number
of
which
is
5
like
pearls,
part of the
lieutenants, 30 sergeants,
consisting of
vests, and
a
the 6
made,
each
containing
4
be
attended
to?
?
(1
where
there
viz.
(k
If
n
be
even,
less
than
u
what
has
to
q
+ 1,
and
coefficient in
question is
in (l
n be
unity, and the sum
of these terms will
chapter
in
r
ways.
Hence
Dr
vacant
The number
which
the
places
r assigned
.
by
writing the
before,
set is called
substitutions
it, and the last by the
first, is
by
the
symbol
(abcdef).*
intervals round
The
decomposition
is
by
no letters
in common,
will
be
§
a
T.
case
of
a
pair
ance
that has
place,
exhausted,
or
until
we
on the left.
Substitutions,
box
marked
1,
2,
how
many
—
be
black,
5
distributed among
to?
(5.)
the
a
set
two
can
be
made
without
be
set
to
m,
the
holes, the
attended
to
and
empty
holes
than
q
letters,
7i]
.
.
be formed
with 2n
How
and no
four concyclic.
Through every
two of
diagonals
is
no
four
these
no
three
are
parallel
or
coincident.
(12.)
curves be in a plane each cutting
all
the
to
be
algebraically
greater
or
less
than
h
as
it
the two
sides of
an inequality
cannot, like
the two
For (P
that ( -
2)
negative,
and
we
>2;
+
the
theorem
x^
x-y, or both
negative.
Hence,
taking
every
possible
(x
+
Hence we
be.
>q{p-q)afl-'{w-lf,
to
be
established.
In
(2)
J[fa
general form
because of
its great
(1
,
See
especially
Schlomilch's
Handbuch
fundamental
limit
theorem
function.
The
use
of
a
it
It
is,
therefore,
suffi-
cient
p,
q,
supposing
a,
6,
c,
Consider
now
m hes
Differential
Calculus
; and
the
elementary
proofs
hitherto
given
have
n real
1
then
{ab
• • •
?i.
?2>
If
a,
h,
c
xyz
is
negative.
(42.)
If
A
the
equation
(1)
and
/(a, b,
brevity
<f>
of
vice versa, then
<l>{x,y,z)='B,
<t>
(x,
y,
By means of the two general theorems just proved, we
can
problems
n (a/A)
.
. .
;
7)
volume
of
the
to the
Let a, b,
x,
y,
z
when
x
put
nj
some of
multipliers
in
a
We
may
write
u
+
^
of
determine a
the 3rd.
to
dominate
the
sign
of
as
above
indicated,
1=
condition
x-y
when PA
sphere the
order
that
a
were
discussed
in
chap.
xv.
there
was
inasmuch
as
it
English
courses
to
calculus
;
The
value
of
{a?
according
as
we
in-
crease
it
down
function
in
chap,
xv.,
a;=o-0
as
we
please
by
sufficiently
diminishing
h.
please
by
sufficiently
diminishing
which it
being a
join the
For example, (x
We
have
when
u
and
v
are
infinite
but
reduce to
critical
values
only one
fundamental case
of indetermination,
indeterminate
form
X
CO.
of
the
.
.
.}.
of
ascending
thus
(see
chap,
v.,
z,
the
the
former
powers
of
x.
Each
of
(^' -l)/(ir-l)
-
«,
§
*
r=l, as
It is
where
understood
{x-y)lylyV-y
r, i.e. that
=
log
{3/2}
Therefore
his
Adnotationes
ad
Eiileri
sometimes
called
Nicolai.
{1/1
limits
of
theorems
are
Now
w
13,
14
cauchy's
theorems
83
II.
L{f{x^-\)-f{x)}^Lf{x)lx,promdedL{f{x+l)-f{x)}
K=«
f(k
it
results
that,
by
/(k
Algebrique
may be well
large
theorem
a>l
Since n
we have
a
finite
>
Next, let m<
the form
by
be
determinate.
We
of
a
tangents
prove the latter
inequality
of
last
para-
graph.
For,
if
*
analytical treatment
from
would
take
the
form
1*,
§
we
X CO
be
deduced
(1),
which
we
lower
deduce
(w
. . . +
L {{a
(2).
b'
Dirichlet
=
the
through
A,
B,
Archimedian rule
a
mathematician,
either
practical
or
scientific.
t
but
difficulties
in
the
Theory
but in
of
his
Lemons
subject
is
to
a
greatest
number
a
and B.
is the
also
continuous,
such
sections
in
a„
might
since
n
may
they
form
a
non-decreasing
or
a
clear
that
by
carrying
Hence
every
convergent
sequence
quantity
e,
however
small,
onefold
of
real
quantity
(S)
built
upon
appropriate
abstract
defini-
tions,
assumed
that,
if
we
choose
any
us
to
started,
we
define
the
limit
of
the
infinite
sequence
of
real
quantities
til,
«2,
small.
satisfies the
definition of
of x,
sa.yf{x),
define
109
regarding
L
a positive
x
+
+
is
= oo.
(cosmx) /^,
x
for
that L
and
when
following
equations*
infinite
the two
summands
and
multiplicands.
difficulties
just
series
are
most
questions
regarding
the
convergence
of
infinite
of
last
chapter,
L
Sn
ofthefwm
3m,
Sm
take
in
order
to
vergency
of
the
series
U1 +
U2
into the
is
obtained
by
applying
the
result
of
originally
con-
vergent,
it
will
remain
so ;
hence since
L
Some
of
But,
by
hypothcsis,
But,
by
hypothcsis,
of
sign,
we
suppose,
they
are
all
positive.
terms
convergent.
all the
tests of
L (n) be finite.
a
positive
=A say.
other
important
cases,
the
Exponential
Series)
is
convergent
as Sv^
convergency or divergency of 2v,„ will be more apparent than
that
limit than the
of the
2 *+^
S„
positive
integer
<j:
2.
%
xf{x)
Grunert's
Archiv,
Bd.
67
(1882)
repeated
application
of
Cauchy's
Condensation
T^
Tr==\{x\xX^X . .
(by
n=oo
(1).
where all the logarithms
In
the
pre-
geo-
metrical
a certain
=
=
criterion
somewhat
troublesome.
We
shall
therefore
if
is
of the series before
within brackets,
M
m(m
oscillating, if
§
periodic
arrangement
was the
of convergent series
principles of the
n increases,
fm+n
then S/(n)
otherwise
divergent.
The
second
step
is
periodic,
series to the
positive
ultimately
vergence
of
the
series,
last series,
(
-
Sl/n a
By
(1)
is
divergent.
It
n
increases,
and
if
i/a„
Hence, by
Un
+
^),
*y2/l
ahsolutehj convergent.
series into
if
/S'^
denote
n=x
retained Cauchy's
original enunciation
latter
case.
SJ=
L
Sn-
Hence
the series
if
we
take
the
most
Cauchy,
Ohm gave
examples of
if
of each series, we
We
may,
to
secure
convergent, have
2
-lx
value,
smaller than
any given
is said to be
restrict
ourselves
to
we
can
for
every
positive
value
of
quantity
e,
If it
or
convergence
to
the
uniform within
p).
Hence,
in
-
+
R.
Let
the
a
z
is
unrestricted
we
can
by
The sum
a
single
valued
function
Lowever
for
all
points
within
R,
Consider
(z)
z
without
divergent.
finite positive
and
value
of
\zq\
the
circle
|2;|>|2;o|;
for, if
it converged
in
divergence, or
that is,
Weierstrass in
his well-known
applies
of
z,
say
(f)
(z),
within
not
If,
please.
same
circumference
of
its
of P
^=0.
At
points
of
convergence
(J?
of
n,
within
every
formed
in
see,
factors
ultimately
become
unity
that
the
most
may
be
left
to the
the
term
convergent
to
less
discussed.
Given,
however,
(1
the ultimate
convergency
of
n
(1
includes the
of
unconditionally
convergent.
The
zero, given
then,
when
c
Argand's
diagram
and
if
integral
function
P„.
when
factors
of
Pn
vanish
be
contiguous.
If
there
be
(1
of
f{z)Qn
by far the
Pn
is
always
+ •
• •}•
simpler.
Let
n
be
(6).
last
paragraph,
(5)
and
(6)
l)
(1-
chapters
to
refer
to
our purposes.
be
limit of this
173
first
converge
to
Ui,
U^,
is clear, however, that
. .
.,
all
the
terms
former
below,
From
this
it
follows,
The last inference holds,
first
sense,
then
each
of
the
horizontal
unaffected.
Hence
S as
by
suffi-
ciently increasing m, make the residue of this series, that is,
the
horizontal line
;
residue
of
S^.,
of
as
many
as
we
and
consequently
deduce the
the
diagonal
series
of'Zu'm,n
—
kind.
Suppose,
for
this
series
^m,n—
~' 'rum-
Hence
there
is
a
 skew
arrangement
array
and
on
the
same
fourth
way.
are
also
For,
if
we
that
is
to
say,
ST'^
is
divergent.
mathematics*.
Babbage,
chap,
xxviii.,
terms in the
the four ways,
its sum is not
as
a
particular
case.
convergent
for
provided
are
the
following
+
...
a<
or
<tl/e.
(Bourguet,
1
(27.)
If
p
of primes
series
positive,
-
and
after
a
certain
where a„
positive constant,
to
also.
+
form
2
(^„+i
integer,
{l
present, we
convergent (by
1,
and
-
order
of
the
+
.
.
. +
contain
a
vast
last
expansion,
when
la
integral,
is
(see
chap,
xxiv.,
a; +^, where
n is
second
the term
as does
.
.
.
+
the expansion
(11.) (3
(n
of
(l
(24.)
If
n
be
/
and
i^
a
p
according
as
n
is
 ?
fractions
CH. XXVII
nKr
curious relation between
elementary
processes
used
in
the
expansion
of
=
ai^'^+^s
particular
powers
of
x,
(1.)
6,
then
(15.)
li
below.
1
roots of
§
when a;
if
p
THEOREM.
first three,
after
If
n
by taking
+
•
• •)•
Hence,
if
.
vanishes.
(8.)
The
coefficient
^
(12.)
(l
i^p
it^
coefficients.
For,
let
a''
be, provided
it exist
at all.
of
integral
value
whatever.
Then,
of the series.
<
shown (see
•
•
•
»«
by
the
elegant
ance,
because
we
of
the
Exponential
Denote
the
or
negative,
for
the
Exponential
Series.
From
(2),
putting
deduce
(8),
J.
1
Jil
where
p/q
—
(3)
the
decimal
places ;
but
We
convergent
so
long
is the
tion
+ .
.
.
4..
(w-3)},
where
^q,
A-^^,
Jr= i°°(n
+
 -cw( -a-r---(«^)- ' -'^-
+
'
' •
(
~
) ~
+
. .
.,
=
(l
+ xy
convergency
.},
Cauchy in his Analyse
+
. .
.}
Thus,
Briggs
the logarithm
computation for
of
same
time
one
the methods
or
the
Introduction
to
4/10^)
(I-S/IO'') .
the
9,
and
the
subject,
give
an
analytical
view
of
where
the
numbers
the tabular difference has
summed
by
be split into
^Sa;7(w
.
. .
+
since
«
in
commutating
its
terms.
APPLICATIONS
TO
1),
in
the
second
x.
(13.)
From
(Trin.
Coll.,
Camb.,
1878.)
(17.)
li
to
the number
constant.
(41.)
the
negative,
then
character.
and is
susceptible of
graph
being
dotted.
The
curves
to
deduced
the
even in
x
by
(1)
and +
qo
to
see
that
3/„
deduce
from
+
therefore, the
wish
6
to
vary
continuously
from
2;-circle
with
Fig.
4
Pj ;
the
lower
in Pj ; the lower
represent
the
actual
analytical
value
of
w
of
z,
belonging
'i/w
and
corresponding
to
any
value
belonging
to
the
principal
=
of the above
that the principal
w
has
circu-
lated
therefore cause
we
have,
in
p
the
spiral.
If
we
z-plane with
inverse function.
(13.)
difficulties
of
this
rests
directly
on
the
fundamental
for
real
values
of
we
deduce
several
important
particular
cases.
For
example,
(6)
and
(7)
give
C08a
in
(4),
(5),
(6),
(7)
identity
leads
to
two
trigonometrical
which
of
Yieta
9
AND
COS
9
277
to
be
remembered,
say
are
form
r
formulae
of
It
by
no
means
numero
terminorum
infinitas,
radian.
of every
(5.)
§
(9)
^{1
to ex-
plore the
centre
greater
than
theorem we
vergent when
owing to
former, and
may
of
course
function,
and
not
2-71828
It
for
^
over
namely,
to
the
right
left
half
of
the
Log 'W
piincipal
value
of
(2).
The
value
given
in
(1)
would
then
be
the
^-th
branch
of
both
a
familiar.
If
we
confine
any
real
quantity,
the
2:-plane
Abel so
resting
it
follows
from
the
exponential
addition
theorem,
namely,
Exp
{izx
+ iz^
positive or negative integer,
Circular Functions have the same
real
periods
odd
function,
zero
sign
to
complex
values
of
the
argument,
might
;
the
relations
connecting
general hyperbolic
+
metnoria
those of
Again,
cosh
X(unh^/^)%1.
as
before,
theory of
equivalent
equations
just
given,
coshyp. and
Q
y
may
the hyperbolic
entitled
m:
(39.)
L
{cosh
a(x+
h)
x^la^-y^lb^=l',
A one
=
R
of the values u
x,
we
y
y-axis
bisects the strip, corresponds
spond
corre-
would
be
along
or
partly
imaginary.
14
of
the
{Tan~^ 'w
W
corresponding
to
the
point
{x,
y)
on
Z,
then
we
have
p
(cos
a
double
the principal branch
+
+
any
will correspond an infinitely
means
of
infinitesimal
detail.
The
propositions
just
stated
third order.
may
be
regarded
theorem
is
are
connected
of the
w^a^Jz
functions, as expounded
of the
equ&tion
each case, where
:
of
the
general
commensurable
values
of
m,
provided
( ') and (2')
^C4-.
_
m{m-2)
It can,
however, be
-
excluded.
(Abel,
sin
d
(cos
d
series
related
to
the
exponential
series
tan~^ is to be
double
limits
sin
61(1
put
9
+
(3.)
Expand
X
from
(1)
a7id
(2)
observe
that,
side
(61).
series in
values
calculate
E^,
E^,
E^,
the
series
involved
convergent.
which
render
the
series
l
(5);
a?Coth;»
(10).
coefficients in the
at a time of
{l
^^^
for
example,
J(4)
r at a
time of x,
r
at
a
x.
ON
THE
Let
IT
p=w
careful
to
that
we
have
p-i
p
purpose
it
were
have simply
as
putting
u
(1).
In
(2),
provided
9^^
(2n
product
on
the
right
of
(3)
(4)
sm\{2n7r
established
is
the
same
M-
sin
the
n
from
OF
TAN
6,
COT
6,
COSEC
6,
AND
SEC
6.
(3),
provided
Q
be
small
enough.
The
two
series
(2)
and
(3)
must
the
convergent
series
of
all
terms
whose
namely
3,
Hence
cr^™
In
this
way
(12),
1/5 - '+'-.
various
appear
approxi-
mate
j
but
only
(1),
since
the
double
series
arising
from
the
expansions
(^>'
(11).
so
that
C
sJ{2Tr),
as may
be easily
*
and determine
the
products
r
at
a
Sum to n
+
of
1/m,
it
cannot
be
said
to
be
more
l)y
of
y
may
make
(1)
an
intelligible
identity,
without
loss
of
generality),
and
+
(5),
(11).
Now
one
tion
(2).
Now,
the
+
,
.
graph that we
.
.
of a; a
is
an
ordinary
synthetic
irrational
algebraic
it has
expansion,
V
involved
each
one
of
them
a
value
thread
of
the
y
corresponding
to
x
a power
positive
value
of
A.
t
It
is
take the
value merely
of
equal
the projections
for simplicity
of illustration,
A=+2,
B=-3,
C=+l,
the type
for
it
with those of the
finite
when
finite
records
left
Newton's
diagram,
and
.
degree than
those already
will
be
observed
=
(27)
belongs
Example
2.
To
find
a
(16)
by
(JJ
first
approximations,
letter
to
Oldenburg;
but
is
(1776).
(See
de
Lagrange,
memoirs on the
1885), p.
Verdnderlichen
Grosse,
for
a
good
function-theory.
The
English
student
from
varioua
of?/.
a;
and
y
being
determined
(F.
82).
(9.)
.
.
.,
regarding the present subject.
. .
.,
all equal, and
primary
series
A''mi,
A^'+'u,,
A^'+'u,,
A^+^Us,
A^+^Uu
A^'+^Ma,
A'^+Swa,
A'^A^Un
form
c '^,
+
differences,
we
have,
in
like
manner,
Newton,
Principia,
lib.
in.,
lemma
v.
(1687)
above theorems
may be
reduce
to
con-
or
is
an
integral
function
of
by
giving
=
figurate
number
of
the
(m
^JBT^, which
all
to
^af^
it
gives
stated
by
terms become
ser es will
the
wth
term
sum to
must
for
its
«th
term
results
of
chap.
XX.,
differences,
we
get
1,
formula
gives
a
 
the
right-
positive
integer,
then
than
the
modulus
of
that
by
the
equation
+
Hence
a
recurring
series
of
If
the coefficients
JJ^—UnX^,
 
§§
generating
function
by
the
process
of
ascending
much more
the sum of
r terms of
sum of
r terms
and
(12.)
If
consecutive
of the
power-series u^ +
be, if the
infinity
+
. .
A;=4, ?ft=3,
identity, we
numerators A, B, .
be reduced
to a
of
series.
It
contains,
'''
y-^
series
converges,
and
m
(aa+2'i)(«2+F2) (ai+i'i)(«2+i'2)(«3+i'3)
iJlPa
obvious that
^Y ,
say
positive
integers,
and
G.C.M. of A
G.C.M.
fractions,
a
being
a
positive
integer.
(18.)
If
a
be
a
positive
integer,
+
hand, the fractions
simplifica-
tion,
as
under)
so
that
their
numerators
and
denominators
are
integral
numbers,
by
pilqi,
p-ilq^,
3
it
holds
piq
into
a
continued
fraction
a^H
9n-r-X
fraction may
and
in
many
cases
convergent
wJiose
corresponding
value
of
the
whole
continued
fraction;
+
1
1
1
^1
+
or, if
same
way.
Since
 
§§
continued fraction.
treatment
of
jfinite
con-
find that, from
have no certainty
the case
when n
each of
1/113
(8.)
If
K
may
be,
(9.)
two
con-
divide a„
relation
connecting
the
numera-
tors
be
prime
to
each
other.
CLOSEST
COMMENSURABLE
of them.
Now
fraction
whose
denominator
does
an increasing or
convergents
of
even
order
with
their
intermediates,
and
form
from the fact that
given
by
Wallis
(see
his
Algebra
(1685),
Descriptio
Automati
Planetarii,
1682).
One
which most
779
approximates
207
to obtain an approximate
moon
and
earth
have
made
respectively
the
'5306
<
above
series;
in
fact,
70
days
every
289
years
would
be
a
difference
of
earth
than
3°
since
lunations,
four
fact, it
obtained
by
taking
higher
convergents.
Exercises
XXX.
(1.)
Napierian
to 1860
so
approximately
at
the
same
time.
(7.)
Along
 
=
+fn+rQn+r-l)
by
Qx'.
For
to
Lagrange.
For
the
details
of
+
(qn-li^n+qn-.T
2R steps
any
one
in
which
Pn
and
Qn
are
not
both
positive.
It
should
be
that
P„
and
Qn
ultimately
become
positive
=
not
in
+
*
«
(3)
and
(4)
may
agree,
so,
we
should
«« is
a
{Pi
JW;
none
of
the
partial
is
true for
and
of
calculating
the
constituents
reducing
surds
{L
1c
and
simple continued
it.
(8.)
and
find,
where
you
can,
closed
q,
q'
quotients
-
sJN
Og
is
satisfied
integral
num-
in
We
are
third.
We
shall
also
treat
the
§
arithmetic
should
first
read
Theory
of
on
§
=
To
get
positive
solutions.
2°.
If
case
:
(3)
will
have
integral
solutions
the
general
form
y
side
occurs
among
the
quantities
integer
have the
will each be + 1. Hence we shall have the system
of solutions
im-
portant
theorem
(1),
we
may
express
of ^C (see
(9)
{xxx-Cyy,y- C{xy,-yx,f
XXXIII
If
it
also
happens
that
in
the
condition
a;i-Cyi
5,
15,
25.
solutions
is
a
and
DEGREE 487
Here
we
??
each
case.
Hence
y
of
9x2
_l2xy
positive integer
J»8.
ways
by
Solve
the
following
systems,
and
which
other
(42.)
x2
continued
fraction
^i
the
demonstrations,
value
of
a
non-terminating
continued
fraction
continuant
of
continuant
are
all
unity,
it
is
usual
to
a simple continuant.
\fl5,.,
po
and
q^
by
K{
a
continuant
for
which
the
system
of
numerators
and
denominators
under
con-
sideration
furnishes
no
and
replacing
by
b^,
Then,'
omitting
memoria
technica,
given
by
its
discoverer
the
two
vertical
(14)
inside
the
;
sign
is
the
number
and
these we
fermat's 499
continued
fractions
has
greater
reverse
order.
Let
p
Jour.,
1855).
-
nth convergent
g„
every
continued
a,
h.
and
natural
manner.
Exercises
XXXIII.
(1.)
Assuming
qj
continued
fraction
in
question
always
converges
to
the
numerically
the
numerically
greatest
root
of
(17.)
Prove
the
following
equivalence
theorem
}
]
exist,
is
for
its
nth
convergent
important
to
notice,
all its
found for testing con-
continually decrease,
while any
convergent,
it
follows
that
Baxjerischen
Akad,
of
:
be
formed
therefrom
by
omitting
the
succession
of
its
=
(2),
then
—
divergent
or
convergent.
following characteristic
the relations
If
we
dismiss
from
our
presently.
CONTINUED FRACTIONS.
n
we may leave
get
bi
(10)
or
(11)
is
di
)
 ~
d^di-e^i-
that
is,
G
by
itself
and for
all positive
{n
CH.
XXXIV
verify,
(Stern,
Gdtt.
Nach.,
1845.)
(12.)
Show
that
these
conditions.
(14.)
Show
that
XXX
* *
with respect
+ r,
and
N-
qm
of
numbers.
one only
whatever, and divide
be a
(A) is
form
Ip,
or
Tpil.
other of
also
^3=
 
consecutive
we
prove
of any
that our
theorem holds
a^+b'^=c^, then
First
account
common
factor
X,
readily
from
the
principles
of
+
.
.
Cor.
1.
Since
0, 1,
no
integral
function
of
x
can
furnish
prime
a; be prime to
a;^
17morl77n-l.
Here
/(0)
17
divisibility
can
be
at
/„_i
{x)
of the form
such that
placing each
digit differently
x^
2* -
3?re.
(36.)
powers cannot be
use
of
these
tables,
factor, namely
,
.
should get
2*
=
of
number
of
integers
(including
1)
which
laitta, 2aia2, 3aaa2>
we
have
laiagfts,
first line is
number,
not
541
For,
since
P
and
Q
are
. .
.
.
.
.
speaking,
(3MiMf^
where
J(
will
be
the
power
in
which
p
occurs
in
ml.
Write
(8.)
Show
by any one of a given
set
and
F
(71)
the
+
then
we
should
have
k +
ra
to m he
;
the
series
0,1.2,.
m
and
before,
are
all
prime
to
cycle
of
m'.
Consider
any
two
terms
say the
the
same
remainder
with
respect
to
w
as
prime
to
a,
are
10,
deduce
Fermat's
Theorem*,
Euler'e
second
-
number of
to m,
be divisible
by m.
But, since
m is
the
series
2,
3,.
. .,{m-2)
in
allied
is of the
3Im +
1.
Hence
.
residues.
cube
same
1.
(5.)
m^-^
.
.,
. .
.).
{pn
-p
+ l)th
(I.) those' in
the
number
of
partitions
of
a
of
n
into
p
number
of
number
of
an
odd
integer;
in
the
series
1,
2,
(7)
(12),
1 1 1 1
37
40
44
4
5
6
9
94 103
141
434
9
By
(9)
. . ad
ao,
(16).
Examplel.
P(20
exceeds
q.
Hence
we
have
the
identity
(19).
(II.)
P{n-p\q-l\1^p)^P{n-q\p-l\1^q) (20).
(III.)
..{l-zofi),
'
therefore,
by
(II.),
=
-2P(?^-/^-jo|*|:}>^)
(24).
Here
the
summations
are
with
come out or negative, this indicates
that the
impossible.
+
with partitions,
magnitude,
and
may be
graph
last
before.
The
new
graph,
therefore,
q
orfewer
parts
the
greatest
of
(27),
a
new
result
Pu {n
placing it
EULER's expansion
CH. XXXV
to the number in the last slope or exceeds it only
by
one,
first
These two
(8.)
universally
true?
(11.)
values
3, .
.
.
.,
= P{n-l\p-l\*)
of the
;
the men
of an
The
alternative
remark
cases by
hazard
played
with
cards,
dice,
&c.
regarding
the
tossing
of
a
the
series
by
however
which are
not apparent
in such
 long run.
Unless an
out of
This
follows
at
arrangements
we
(Wp/^n);
(2 ~i-
l)/(2 -l),
Example
6.
can take
rook
on
any
of
these
8
and
row of
rook
can
42/62
rook. If
he is
by
9000
is
most probable
throw with
the two
+
odd
suppose
the
m
odd
integers
written
even integers into
the spaces so that there shall always be one at least
in
every
to
are of such
no more
reasoning would
lose all
calculate
the
probability
that
the
throw
2, 3,
5/36
respectively.
dice does not exceed 8
is
(l
of events
have
happened,
and
so
on.
It
must
be
observed,
:
other does
now
not
specified,
second does
respective probabilities
white
ball
is
p.
a
white
ball
is
there-
fore
(iJ
by mixing
the chance
of drawing
suppose
an
ordinary
done
at
least
once.
The
are
1/2^,
1/3^, . . .
1/n^,
1st,
exactly
m;
2nd,
exactly
m
(sec
chap.
XX.,
2)pqpqq
are
as
many
ways
of
n
the
probability
in
question
is
expansion
of
{p
these
probabilities
ought
throws
1,
2,
probability
2?
is
in
integer,
and
8
in question
cases.
matter in this
made
(Pascal's
Problem.)*
The
issue
in
(1').
We
might
adopt
independent events
arise from
from
Spi
{p-zPs
+PiPi
of
—
. .
(
—
n
events
happen
is
therefore
^PlP2.
great historical
his
Consider
A'&
chance
lost, or
drawn), we
(1)
that,
if
we
Mq
particular case
the players is unequal
disparity of
shown
;
when there are
n cards ; so
drawing
6
chance
13
a
named
the
r-permutation
of
n
letters,
9,
throw at least
two sixes with
stack is
if any other number the game is drawn.
Show
that
is decided
3,
yard
apart.
is
respective chances of winning
he
loses
tail,
he
might
give
various
estimates,
according
as
get
nothing
the
same
be seen that
a
lottery.
he
ace the second throw;
20/6,
20
expectation
is
therefore
¥{'4HHyKAy-
2a,pi (1
both of the
other, but not
would still
for such
indicated
by
the
numbers
each
year
is
uniform
§
year
is
H^m+Wl)+MWl
the
intervals
between
the
mutually
exclusive
contingencies
through
the
year,
if
A
is as
is
J.
The
chance
of
^^^
Hence the
of the
{^<^l
the table
599
the
first
payment
the use
tlie
upon to pay
saying that
the annuity
concerns
the
definition
be continued
(5)
\J)
as
the
reader
Im
j
is
living
r
years
consideration
that,
much for
respective nominees
table
of
annuities
annuities.
3rd
column
from
are
Montmort's
Essai
d'
Exercises XL.
(2.)
1st,
of
coin
value
of
the
;
up
again,
marked
2,
S, if
opponent
are
1315'/256
and
125S/256
respectively.
(Montmort.
See
expectation.
(8.)
A
man
is
(10.)
A
last
of ages
alive,
{n /(Jn) }x /^;
+
r be odd.
if
n<(:
3,
the
Ist.
(19.)
{2
+ 3x)l(l-7x
(l
14,
1,
1,
5.
(4.)
31,
1,
1,
3,
5,
1,
2.
(6.)
8,
243,
5, 2;
0,1*2,13,
8,12,12,8,13,1*2;
1*2,
5, 7,20,
(a6
(11.)
2(r-
n+l
ively.
(3.)
£8
: 5
184, 287
142,
150,
171,