PKOFESSOU G. H. DARAVIX OX ELLIPSOtDAl. HARMONIC ANALYSIS. 
541 
iiW 
: A 
'oos^ d 
i'cus- 6 
<«=/(! )[l+f/3+ Hf/3'|. 
^ fW=/(2)[l-i/3-';L/3-^J. 
/3 I (4/i- + l)/3“ 
1— T 
8(u —1 )(m —2) 
, »> 2 . 
cos-()A-?e=/(2)[l+i/3+-U^^. 
cos- 0ArW=/(u+l) 27,-8.„(„_1) 
y6> 1 
(yi). 
Before using these for tlie determinatioii of L, M, N, it is \vell to obtain one other 
result. 
We have seen in (82) that 
i-s ;, 
23'V + «I(r !)V-«! 
COS''' 6. 
i +si 
Therefore (PfdiJ. = t—^ 1 (- )'+*/ (r +1) — 
. ^ 1 t- o ; g -j / 
ir ; 
+ s!(r !)b'-sr 
t + *' ; 
Hut this iuteoral is eciual to •:-;; therefore 
^ ^ 2iAl i—sl 
M 2~'-r + slirlfr-sl 27 + 1' 
Hutting and 0, and comparing with the values of in (82), we have 
Sa,,/(,■+!) = ,1^1- 
- 72 ,- 2 /('^*+ 1 ) — n: 
i + 1 : 
2U1 : 
(92). 
^21. Liteyrah o f the squares of harmonics ivhen s = 1 amd s — 0. 
In (83) w'e liave 
-:=V 
72,-2 
■^/■(cos''"''' 0/Bf cos^'' 6) dO. 
t 1 J -^TT 
. 2/- + J . 
Therefore, noting that ff')-=—^-ff'-\-\f and using the integrals (hi), 
L 
2 —- 72 ,- 2 //’+ 1 ) 
I _._ (4/' + o)/j" , 2/'+ 1 0(1 i^_ 
2?'"^ 2r 2(r-l)/- 
+ 7o/(^)[l + 
