PEOFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
529 
Again, omitting intermediate steps, 
-I 
/ 
da 
, {fi^(.y + -i) + T(W^'(55j~ + 304; + 756) 
'2i +1 i- 
-.rMj^[i/?+i/3’(2,; + 7)] 
0 '' 
la 
2 i + 1 i ■ 
~7rMj^[|/3 + i^^(^+l)] 
(74). 
(74). 
^ 17. Portion of the Tntcr/rntiori. in the case of s = 0. 
We are to find pi [(P/)' —(P,)'] dcr, leaving two integrals of the tvpe 
f P (P>)'^^^ determined subsecpiently. 
It is only necessary to consider (li;, since the other case is determinable from it by 
multiplication to found in (38) of § 9. 
Following the procedure of the case where s=l, we have 
j ¥,,W = - 2fef PPJ.VI [<?,PP ‘+i (<ft)HP^)T < V. 
“ ^^ (^ -1) (^- -) (^- 3) +/3' (7o)'(^+1) + 2) i{i -1 )]• 
Then since q. = h <h=ils^ 
|F,«7^=--|-)J/3[y+l-(2i+l)] +J;/3^’[5/+14/+12-4(2,'+l)(,/+3)]j. 
Now (C)"=I + 2^j9. cos 2cf,+l3- [(2p^+^(p,,)') cos 4(/)+l (p.)-], 
and as far as material 
('Cf P 
(T-P cos 2(^)i “ ^+^'[te+i-(P 2 )“+A] + ^(i-+22P) cos '2(f), 
= )l + -f273-(/'-p‘d-f>): {l-P(y-l) cos 2(^], . . (75), 
yp=-4i ({+!)=-A/. 
Since 
O 
•_> 1 
VOL. rxccu.—A 
