502 PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
The coefficient of/3“ in tlie expression for is 
~ (i + s) (i + ■$ — 1) ( i + s — 2){i s — 3) {i - s + 1) (i — $ + 2) (i — s + 3) (i — s + 4) 
'L (.5-1)(.s- 2) (s-l)(.s_2) 
( i — s)(?: — s — 1) (i — s — 2){i - s — 3) (?: + ■$+ 1) {i + s + 2) (t + .s + 3) (i + -s + 4) 
(s+1)(.5 + 2 ) (s+l)(s + 2) 
2(^' + s){i - 1 - -s — l)ii — g + l)('t — .s + 2) 2(t — g)(i — .9 — 1)(-^ + s + 1)(^ + -s + 2) 
(s-lT (.s+1)-' 
2{i + -s — l)(j' + s){i + s + l)(j: + s + 2) '2{i — s — l){i — s) {i — s + 1) {i — -s + 2) 1 
- (.5 4-1) “ (.s_l)(.5+l) j- 
^ ^ , s _ -4 
In the ex})ression for E'' the first jDair of these terms are multiplied by —^— ; the 
second pair by 
and the last pair by 
The reduction of terms such as these will occur frequently hereafter, and I wiU 
therefore say a word on the most convenient way of carrying it out. It is obvious 
that the coefficient of ^ may be arranged in the form 
Ki{i + 1) + B(2f + I) + C. 
T1 le coefficient A is equal to the coefficient of in the original expression, and if 
we put f = 0 we have B + C, and with i = — I, — B + C. Hence A, B, C may be 
easily determined. 
Again the coefficient of may be arranged in the form 
Ar (f + 1)2 + B (2f + 1) f (f + 1) + Ci (f + 1) + D (2f + 1) + E. 
This may be written 
Ai4 + 2 (A + B) + (A + 3B + 0) r + (B + C + H) f + D + E. 
It is easy to }nck out the coefficients of i^, and we thus obtain A, B, C. 
'flien putting i successively equal to 0 and — 1 we have D -f" E and — I) + E. 
In order to express the results succinctly I use as before the notation 
V» — — (f - + i)B' + 2) _ 
s2 - 1 ’ ' s2 - 4 
and I usually omit the superscript and subscript s and i. 
