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The properties in question are as follows:
If i and m be unequal positive integers,

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The following is a proof of the first property.

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2

Multiplying the first of these equations by Pm, the second by P, subtracting and integrating, we get

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+ {i' (i + 1) − m (m + 1)} [P ̧P„dμ = 0.

Hence, transforming the first two integrals by integration by parts, and remarking that

we get

i (i + 1) − m (m + 1) = (i − m) (i + m + 1),

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or

i

m

+ (i − m) (i + m + 1) [P ̧P„dμ = 0,

(1 − μ") ( P2 dP ; — P, P) + (i − m) (i + m + 1) [P,P_du=0,

m

άμ

-P、

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since the second term vanishes identically.

F. H.

2

Hence, taking the integral between the limits

1 and

+1, we remark that the factor 1-3 vanishes at both limits, and therefore, except when i— m, or i+m+1=0,

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We may remark, also that we have in general

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dP dP
P

Pm du

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i αμ

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a result which will be useful hereafter.

11. We will now consider the cases in which

i-m, or i+m+1= 0.

We see that i+m+1 cannot be equal to 0, if i and m are both positive integers. Hence we need only discuss the case in which m=i. We may remark, however, that since

P1 = P_(4) the determination of the value of Pd will also

P=P_

-(i+1)9

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‚: (1 − 2μh+h3) ̄1 = (P。+P ̧h+

2

... + Ph' + ...)*

=P2 + P2h2 + ... + P2h2 + ...

+2P ̧P ̧h+2P ̧Ph3 + ... + 2P ̧P2h3 +.....

Integrate both sides with respect to μ; then since

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we get, taking this integral between the limits - 1 and +1,

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all the other terms vanishing, by the theorem just proved.

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12. From the equation [P.Pd = 0, combined with

-1

the fact that, when μ = 1, P ̧=1, and that P, is a rational integral function of u, of the degree i, P, may be expressed in a series by the following method.

We may observe in the first place that, if m be any

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For as Pm Pm-1... may all be expressed as rational integral functions of μ, of the degrees m, m-1... respectively, it follows that " will be a linear function of P and zonal harmonics of lower orders, μ of P, and zonal harmonics of

m

m-1

m-1

lower orders, and so on. Hence "Pdu will be the sum of

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a series of multiples of quantities of the form PPdμ‚

1

m being less than i, and therefore "P=0, if m be any integer less than i.

Again, since

-1

(1 − 2μh + h2)−1 – P ̧+P ̧h+ ... + P ̧h2 + ... it follows, writing - h for h, that

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

P ̧−P ̧h+

And writingμ for μ in the first equation,

...

(1 + 2μh + h3) ̄* = P' + P2h + ... + P'h' + ... P', P... P.. denoting the values which P., respectively assume, when -μ is written for u. PP or -P, according as i is P involves only odd, or only even, as i is odd or even*. Assume then

even or odd.

1

P,... P1,
Hence
That is,

powers of i, according

P1 = '‚μ3 + A¡-¿ μ1? +...

Our object is to determine A,, A-9....

i-2 i-4

Then, multiplying successively by μ, μ,... and integrating from 1 to +1, we obtain the following system of

equations:

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2i-1 2i-3

2i-2s -1

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And lastly, since P1 = 1, when μ = 1,

A+ A12 + ... + A12+ ..... = 1;
A+ Ai-2+

the last terms of the first members of these several equa

tions being

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13. The mode of solving the class of systems of equations to which this system belongs will be best seen by considering a particular example.

d'

αμι

* This is also evident, from the fact that P, is a constant multiple of (u2 - 1)'.

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From this system of equations we deduce the following, O being any quantity whatever,

+

y

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1 (0-a) (0-ẞ) (a+w) (b+w) (c+w)

+

a + Ꮎ b+0 c+0 w (w−a) (w−ß) (a+0) (b+0) (c+0)*

For this expression is of -1 dimension in a, b, c, a, B, Y, e, w; it vanishes when = a, or 0=ß, and for no other

finite value of 0, and it becomes

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1

==

W

,

when e = w.

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1 (a + a) (a + B) (a + w) (b + w) (c + w)

w (a - b) (a−c)

with similar values for y and z.

(w-a) (w-B)

?

And, if o be infinitely great, in which case the last equation assumes the form x+y+2=1, we have

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with similar values for y and z.

14. Now consider the general system

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