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

The following is a proof of the first property.
We have

dPill

+i (i+1) P;=0, du

dul d

+m (m +1) Pm=0. du

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

d
1 )

du
du

dus

{(1 – wey

) e

(1 – wed)

2

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

i(i+1) - m (m +1)= (i – m) (i + m + 1), we get dP

dP Pm) 1 ) (

du du du du

dP

dP du

du 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 - u vanishes at both limits, and therefore, except when i - m, or i + m + 1 = 0,

dP,

m

Pi

m

P=P

of | r; du will also SP

-(i+1)

-(i+1)

-1

We may remark also that we have in general

dP. P.

du du

(i-m) (i+m+1) a result which will be useful hereafter, 11. We will now consider the cases in which

¿ – m, or ¿ + 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

the determination of the value of give the value of S,P.Pd.

Р. du.
The value of

off",pidu may be calculated as follows:
(1 – 2uh + ho, * = Pc+Ph+ ... + PK + ...;

+ P.h'
.: (1 - 2uh + h) 2 = (P + P h + +P;h' + ...)?

=P+ P.,?? + ... +Pik" + ...
P

Pifh'
+2P.P/h +2P.P.) + ... +2P,P.1* + ...

*
Integrate both sides with respect to ; then since

1
$1 – +19"
(1 – 24h + h) du log (1 – 24h + ho),

2h we get, taking this integral between the limits – 1 and + 1,

h log h

L Pidu 1=, Pdu*

-h all the other terms vanishing, by the theorem just pioved.

-1

=

0

0

2i

'

-1

2i+1

1+h Now log

1 h

2 h

hi

+ 3

+

+ 21 +1

h? Hence 2(1+ +

3

+

2i +1+...)

h?

log **-? (+

...... 2 (1+ - Pu+a", p?d4 + ... + L', + + ** *, pidu +..

L., = 12. From the equation | P.Pudp= 0, combined with

-1

Hence, equating coefficients of ha,

2 Pidu =

2i +1'

m

-1

1

ท

-1

1

1

-1

the fact that, when p=1, P1=1, and that Pi 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 integer less than i, s

il
,

uim Pidu = 0. For as Pm, Pm-, ... may all be expressed as rational integral functions of p, of the degrees m, m-1... respectively, it follows that rem will be a linear function of Pm and zonal harmonics of lower orders, fum- of Prm, and zonal harmonics of

, a series of multiples of quantities of the form (P.Pidp, m being less than i, and therefore 1,m

pe" P.,du= 0, if m be any integer less than i. Again, since (1 – 2uh + ho)"}= P.+P/h + ... +Pik+...

it follows, writing - h for h, that (1 + 2uh +h)-=P.-Ph+ ... + (-1)'PK + ...

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

1

-2

And writing - u for p in the first equation,

(1+2+h+h)-3= P0'+P,'h + + Pidhi +. P', P,' ... P' ... denoting the values which P., P,... P.,

Pi respectively assume, when - M is written for jo Hence

P = P, or - Pi, according as i is even or odd. That is,
P; involves only odd, or only even, powers of i, according
as į is odd or even*,
Assume then

P,= Aix' + Ai-Mi? +...
Our object is to determine A,, Ai-g...

Then, multiplying successively by hi-?, uts, ... and integrating from - 1 to +1, we obtain the following system of equations:

A
24-1 21 - 3

2i - 2s -1

A+

+ 2i - 3 2i - 5 21 - 2s 3

2

Ai-2

Ai-23

+

+

+

Ai

Ai-2

+

+

- 0,

...

And lastly, since Pi=1, when j = 1,

A:+Ang+...+ 41-+...=1; the last terms of the first members of these several equations being

A

4., 4., if i be even,
¿-1' 3-3

A,
1-3** 1
A,

A 2-2 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.

* This is also evident, from the fact that Pi is a constant multiple of :(-1).

+

w

y

Z
+
+

0,
a + ß'b+ß C+
X
Y

1
+
a to
btw

cto From this system of equations we deduce the following, e being any quantity whatever, y

1 (0-a) (0-1) (a+w) (6+w) (c+w) + + b+o'c+ 0

w (w-a)(w-B)(a+) (6+0)(c+0). For this expression is of – 1 dimension in a, b, c, d, B, Y, 0, w; it vanishes when 0=0, or 0=B, and for no other

1 finite value of 0, and it becomes when e:

ato

C

=

We hence obtain y

1 (0–a) (0-1) (a+w)(b+w)(c+w) +0 c+)

w w-a)(w-B) (6+0) (c+0)

( and therefore, putting 0=-a,

,
1 (a +a) (a + b)(a +w) (6 +w) (c+w)

w (a - b) (a-c) (w-a) (w –B) with similar values for y and z.

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

(a + c)(a +B)

(a - b) (a -c)' with similar values for y and z.

14. Now consider the general system

X=

X;
+

+
a, tama antara