involves but one arbitrary constant, and that as a factor. We shall henceforth denote by Pi, or Pi(u), that particular form of the integral which assumes the value unity when u is put equal to unity.

We shall next prove the following important proposition.

If h be less than unity, and if (1 – 24h + ho) be expanded in a series proceeding by ascending powers of h, the coefficient of h' will be P, Or, (1 – 2uh + ha-1 = P,+P,h+ ... +P,h' + ... We shall

prove this by shewing that, if H be written for (1 – 2uh + h*)), H will satisfy the differential equation

d

dH ď"
du
For, since H=(1 – 2mh + k?) },

1

1 – 24h + h’; H

{(1 – way

ر

dufth

dhe (hH) = 0.

=

..

dH
(1 )
dus

dH
=- 2p HR +3 (1 – M?) H?

du =-2 H +3 (1 - ?) hHo,

1 dH And

Hdh

=μ-h, d

dH

1 hdH .. (hH)= H +h H dh

dh HTH dh

H
=H* {1 - 2uh + h’ + h (e – h)}
H* (1-0);

...

2

dus

dhi (NH)

(hH)=

( {11 (1 -uh)}
un'
dh

dH
3 (1 - x) Ho - μH"

dh
= 3 (1 – μι) Η(μ - h) - μH".
1 d

dH) d? Hence

+
(1 - M*)
rdul

hH
= - 3uH+ 3 {(1 – u?) h + (1 - uh) (u – h)} II"

- 3uHo +3 {u (1+i*)– 24°l} Ho
=-3u (H* - (1 - 2uh + ) Il"}
= 0, since 1- 2uh + h' = H?.

ан)
Therefore

th (1H)=0. dul dh

=

=

This may

also be shewn as follows. If x, y, z be the co-ordinates of any point, ' the distance of a fixed point, situated on the axis of 2, from the origin, and R be the distance between these points, we know that,

RR = x + y + (z' — 2)",

2

Now, transform these expressions to polar co-ordinates, by writing

x=r sin cos , y=rsin O sind, 2 = r cos 0, and we get

RP =pl-2z'r cos +2",

(2

or, putting cos 0 = j,

?
d

d
dr R

du
Now, putting r=z'h, we see that
R?

1
| ho – 24h +1

H2

1 H Н or

R=
R

r

z' dhe (h77).

adhi (hH) +

= ?

h đ and

dr2 R .: the above equation becomes h d? d

d H 1

, du

du
d

dH
h
+

= 0.
dh?

du due) 4. Having established this proposition, we may proceed as follows: If p; be the coefficient of hi in the expansion of H,

H1p,h

H=1+ 2+ + 2*+...+ pl*+...
:: hH=h+p,h* +p,h* + ... +p, *** + ...
72

(hH)=1.2p,h+2.3ph +. +ii+1)p;h' +...

ď (hFI)

or

+

..h

dh

Also, the coefficient of k' in the expansion of
d

dpil

dus Hence equating to zero the coefficient of k', d

dpil

+i (i+1) p;=0. du

Also Pi

a

=

0

P
P.+

h

1

+

+ ...),

is a rational integral function of Mo And, when j = 1, H= (1 – 2h+h)-}

1+h+ to +...+h+...
Or when y=1, p.= 1.
Therefore p, is what we have already denoted by P..
We have thus shewn that, if h be less than 1,

(1 – 2uh +h*) * = P +Ph + ... +Pik' + ...
If h be greater than 1, this series becomes divergent.
But we may write

M
( – 24h + 1) = (1-8+)
2 -}

h h*

= (P. + 1 since is less than 1, h P.

P ++

+ ... + minit Hence P. is also the coefficient of h-(i+1) in the expan

1 sion of (1 – 2uh + h) in ascending powers of + when h is

h greater than 1. We may express this in a notation which is

. strictly continuous, by saying that

P,=P-(i+1)= This might have been anticipated, from the fact that the fundamental differential equation for Pi is unaltered if - (i+1) be written in place of i; for the only way in which i' appears in that equation is in the coefficient of P:, which is i (i + 1). Writing - (i + 1) in place of i, this becomes - (i+1)(-(i+1) + 1} or (i+1) i, and is therefore unaltered.

P,

Then

-1)! 8" (2)

idzi (

and let k be any quantity less than r.

{x' + y +(z— k)?}}= f (z – k), and, developing by Taylor's Theorem, the coefficient of k' is

1 ď
or (-1)"
1.2...i'

1.2...ide
Also {m* + y2 + (z– k)"}-} = (p2 – 2k2 + k*)

1

k k2, }
+

22 since z=

= ur, in the expansion of which, the coefficient of k' is

P.

= }(1-2

P.=(-1' 1.2 ... i dz

Equating these results, we get

poti di
) ().

i The value of P. might be calculated, either by expanding (1 – 2uk + ha) + by the Binomial Theorem, or by effecting the differentiations in the expression (–11',),

gotti
) C

idzi

1.2.3 and in the result putting = M. Both these methods how

ever would be somewhat laborious; we proceed therefore to investigate more convenient expressions,

+1

r