PPvOFESSOE Cr. IT. DAPvWTX OX ELLIPSOIDAL HAPMOXIG AXALYSIS. 
522 
(I — /3cos2^)“% and with accented p’s we ninltiply bv (I — /S cos '2(f))', and we shall 
then have the functions denoted above by K~. 
The function K~ has to be innlti])lied l)y a function of tbe form A + B /3 cos 2(f). 
and integrated from (f) = 2rr to 0. It follows that the only terms in K'- which will 
not vanish are those independent of (f> and those in cos 2(f ); moreover, the latter terms 
are onlv required as far as the first power of 
Now { \ — /3 cos 2(f))~-^ = 1 + I/S cos 2(f> + (1 -f* cos 4(^), 
( I — /3c()s 2r/))- = I — l/Scos '2(f) — (1 + cos 4<i) . 
Then omitting terms which will vanish on integration 
((IT; or 
(\ — ft cos 'l(f)V 
— f> fl + /S' [(/),_..)’ + (N" ! 2 )' + hp<--) + ■ 9 /^" +2 + T'Vlf 
d" l^iPi-'i 4" 2 b + 2 + 5 ) cos 2 
: C; or S/)^ _ , . 
(1 — ft cos 2(f)Y 
— Til + 
+ ft ip '),-2 + 2^1 +2 - ^) G <^^ 2 ( f >. 
However, the latter formula is not needed except for verification. l)ecanse it will he 
derivable from tlie former by multiplication Iw • 
Now if we substitute for the ^fs their values as given in (27), § 8 , we find 
BL'/ or 
0 — ft cos 20)= 
. = T{1 +i|o/8'[X' + 4L + 6 +S'(X' - 2L + 1)]} + lft(l + I) cos 20. 
And multiplying by 
(D/T 
or 
1 — 4 " 8 ), or developing directly 
(C/ or sn- 
(1 — ft cos 20)= 
'fhe.se represent the K'- of our integrals. 
T{ L 4- go/S' P' - (i + - 2 L + 1)] ' + \ft{l + 1) cos 20 . 
'fhen 
27rM(1 - ftf _ l±pl M „ 1 V __ n 
2/ 41 1 - s! ' 
+ - 6^ + 6 - 5'(X~ + 2L - 1) + 2T]} 
X [1 4- 4- 4- G + S-(X- - 2L + 1)]} 
+ IttMA -/Sb 
. 1. L±£; (s + 1) _ ,30.1. L±±| + 1) 
•S / -N i S ( - .S , 
