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 μ is put equal to unity.

We shall next prove the following important proposition.

If h be less than unity, and if (1 − 2μh+h3)-1 be expanded in a series proceeding by ascending powers of h, the coefficient of h' will be P.

Or,

(1 − 2μh+h2)-1 = P ̧ + P ̧h + ... + Ph' + ...

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

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

1

= − 3μH3 + 3 {μ (1 + h2) — 2μ3h} H3

==

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,

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

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

If p, be the coefficient of h' in the expansion of H,

.. hH=h+ph2+ph3 +...+ph++...

:.

(hH)=1.2p,h+2.3p.h2 + ... + i (i + 1) p;h' + ...

Also, the coefficient of 7' in the expansion of

Hence equating to zero the coefficient of h',

Also Pi

is a rational integral function of μ.

And, when μ=1, H= (1 − 2h + h2)−1

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

Or when μ= 1, p1 = 1.

Therefore p, is what we have already denoted by P.

We have thus shewn that, if h be less than 1,

(1 − 2μh + h2)-1 = 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 expan

sion of (1 − 2μh+h)

1

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

P1 = 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.

i

Then

{x2 + y2 + (z − k)2}-1 — ƒ (z — k),

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

Also {x2+ y2+(z - k)2}~} — (r2 — 2kz + k2)−4

since z = μr,

=

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

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

differentiations in the expression (-1)* 1.2.3.id)

ช

dzir

and in the result putting

r

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