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

We shall next prove the following important proposition.

If h be less than unity, and if (1 – 2μh+h) be expanded in a series proceeding by ascending powers of h, the coefficient of h' will be P1.^

We shall prove this by shewing that, if H be written for (1 − 2μh + h2)−2, H will satisfy the differential equation

=-3μH3 + 3 {(1 − μ2) h + (1 − μh) (μ — h)} II3

This may also be shewn as follows.

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

R2 = x2 + y2 + (z' — z)3,

and that

= 0.

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

x=r sin cos p, y=rsin 0 sind, z=r cos 0,

4. Having established this proposition, we may proceed as follows:

If p, be the coefficient of h' in the expansion of H,
H=1+ph+p2h2 + ... + p,h' + ...

:. hH=h+p ̧h2 +ph3 +...+p,h**1+...

(hH)=1.2p,h+2. 3ph2 + ... + i (i + 1) p,h' + ...

Also, the coefficient of 7' in the expansion of

dμ { (1 — μ3) dp.)

dH is μ2) αμς

Hence equating to zero the coefficient of h',

Also p, is a rational integral function of μ.

And, when μ=1, HI= (1 − 2h + h2)−1}

Or when μ= 1, p1 = 1.

=1+h+h2 + ... + h' + ...

Therefore p, is what we have already denoted by P.
We have thus shewn that, if h be less than 1,

(1 − 2,μh + h3)* = P ̧ + P ̧h + ... + P ̧‚h' + ...

If h be greater than 1, this series becomes divergent.

Hence P is also the coefficient of h−(i+1) in the expan1

sion of (1 − 2μh+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)•

i

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

Then

{x2 + y2 + ( z − 1)2} -* = ƒ (z — k),

and, developing by Taylor's Theorem, the coefficient of k' is

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

The value of P. might be calculated, either by expanding (1 − 2μk + h2)−1 by the Binomial Theorem, or by effecting the

differentiations in the expression (-1)*

1.2.3 idz

...

and in the result putting μ. Both these methods how

r

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