PROFESSOR O. H DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
491 
When 5=1, 
= 16“+ + 1) “ ^ + 1))> with upper sign for cosines 
(EOC, OOC) and lower sign for sines (OOS, EOS). 
(Lh = tysSts = yis for all cases. 
But for 0( )S, EOS Ave use the p form, and for EOO, OOC the P form ; and for 
•S ^ -j- -4- 
the latter-^ = 3,-= .5. 
•s s 
Therefore for OOS, EOS (sines) 
= T4l(l - 1-6 + 1) )> ‘h = T(T8 . ( 
and for EOC, OOC (cosines) ... . . (29). 
9.Z ~ 1% S" iV d" 1))) — tIy ^ 
For OOC, OOS, with upper sign for cosine and lower sign for sine, 
Pi — iV {^■> "1 i ”1~ l) ] ) p5 ~ TSF {l 4 . . (29). 
For EOC, EOS, Avith upper sign for cosine and loAA^er sign for sine. 
Pi = - 1-13 {h !■ [1 ± 1-6('^‘ + 1) ] , iC = y¥8 {h 3] (f, 5} . . (29). 
When 5 = 2 the coefficients may be derii-ed from the general fornuda. 
Wlien s = 3 
P = 
- Hl 3} {i2} 
— ~ iV \ L 2 ( ) f, 3 j- [1 ib iV^^' ('- + 1) ] , 
8 + ^^i{i + 1) 
the upper sign applying to cosines (OOC, EOC) the loAA^er t(3 sines (OOS, EOS); 
*h — i'i ’ 9'? — 15^6 0 
But for OOS, Eos the form applies, and for OOC, EOC the P form applies. 
•s- — 2 
Also AAuth s = 3, - — - 
.5 
■S- + 2 
3 5 
5. 
3 > 
8 + 4 
3 R 2 
