PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
495 
Therefore 
|g.j = [1 ~ i/3 cos 2^ — 1^/8-(I -j- cos 4(/))]|^°%f/> 
+ /8 
* “ 'jsin ~ P ^ + (s + -) ^ 
[I — i/8 cos 2(f)] 
+ ^'Ys — 4)</) + ^V*+4|g-jj(’'>‘ +4)^ 
This expression, when developed, must lead to (fD'' or multiplied by a constant 
factor. 
Let 
_ 7).]C^ 
- 14, ^ g . 
(34). 
cos 
Then D/ or D' may be found by considering only the coefficient of S(j). 
Hence 
JT - 1 - i‘o/3^ - - Wp's 
But 
Therefore 
' _ - 1 } 
- ~~ 8(.s - 1)’ 
/>»+2 = 
— {i, s + 2} 
8(.s + 1) 
^>.s-2 + P's+ J. = i (- + 2)- 
^ + y). j 
-*-'i I 
(dV)5 = 1 - ii84S + 3) J 
and 
. (35). 
Tlie reciprocals may clearly be written down at once. 
There are no tactors by which p®, p", can be coiu^erted into P'b P’b P^ ; but 
this is not true of the cosine and sine functions. 
In the case of *• = 3, it will be found that the general formula, holds go(Kl for the 
factor Avhereby SP are convertible into C'^, S'i 
When s = 2, 
{s2 = 2 iJs/3-(l +cos4(/))]|“y./. + - 4/3oos2</.)|i 
+ Dp'Al - i$cos- 24 ,) {“G,/. + 
Then 
= [1 - (#0 + h'^'o + ip\) Y] cos 2.^ -f ... 
S" = [1 — (aV + i'p'i) /3’] sin 2(i + ... 
