PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
503 
Proceeding in this way I find 
i + s! 
= (-)' {i - i^(s - i) + AAE- + 23 - 1 ) 
i + s 
+ 3 (2® — 2S + 2 ) + 2T] 
= (-)' ;-5 {1 + i/3(S + 3) + (3S= - 2S + 1) 
— — 26S — 42) — 2T] } 
(41). 
These results may he verified, for if we nudtiply by 
as given in (33), we ought 
to find ; and this is so. 
The formulae apparently fail when 6’ = 0, I, 2, 3; but when .s- = 3 they still 
hold good because, as remarked above, the general formula for s- = 3 gives correct 
results when properly interpreted. Thus it only remains to consider s = 0 , 1 , 2 . 
When 5 = 2 the coefficients of j3 remain as in (41). In the coefficients of j8~ 
— i - 0) - S {^5 [h ) 9s + 2 
<2 s-i = 0) 2 s-2 — 0? ^^•s + 2 
8.35 + i — 128.3.4 
5 7 +1 ~ ttI'tt • 
In the expression for the coefficient of /3^ inside the bracket is 
[3 {i - 2) (i - 3) (i - 4) (i _ 5) + 8 (i + 3) (i + 4) (» + 5) {i + 6 ) 
+ 72 (»■ - 1) ; (i + 1) (i + 2) + 8 (7 - 2) (i - 3) (l + 3) {i + 4) 
- 24 (i + 1) (,: + 2) {i + 3) (»■ + 4) - 24 {i - 3) (i' - 2) {i - 1) i] . (42). 
Effecting the reduction and writing 12 for {i 1), we find 
4 + 2 ! 
<£: = [1 - - 1) + - 1302 + 80)} 
(43). 
The coefficient of for may be got from (42) thus :—Multiply the first and 
second terms by 3, erase the third, fifth, and sixth terms, and multiply the fourtli 
term by 4. 
Effectino; the reduction we find 
+ 3) + ^i5^a25V + 1862 + 368)} . .(44). 
Observe that there is no factor by which can be converted into P', so that this 
case cannot be verified like the general one. 
