PROFESSOR O. TT. DARWIN ON ELLIPSOIDAL IIAParONIC ANALYSIS. 
;YI7 
But we liave already found in (77) tlie value of j p [(^i^/)' — (?;)'] (hr, and on 
adding it to the last result tlie term independent of l/(2i4-l) disa])pears, and we 
liave 
(p, cr [1+i/5;+f,v,8'M 
. . (%). 
'file square of the factor wlierehy (iTi is converted into C, Avas found in (38), 
namely, 
|-i + py+^K^i(y:_2/_24)] , , , 
• — i*, 77 ^ I 
Tlmse are the last of the required integrals. 
§ 22. TahJe o f hitegrah of squares o f harmonics.''^ 
In this section the results obtained in (71), (72), (1)4), (Di>), ami (90) ai'c 
collected. 
* Aftei' having' completed the evaluation of all those integrals, I found that they may he evaluated 
very shortly hy means of the factors (Jr and E of (48), § 10. 
I find that for nil values of s (writing the eight forms in a single formula). 
r r (I9pr ^ or (Crh'^ 4.M . f e-’ ^ r 
J^ I(Pp)- or (spr 2rn ^ ’ ■ i ep- t 
const, part of 
(Cp or Cr" or^vWiySp)- 
(1 - /i cos 2</))- 
I leave the I'cadcr to verify that this is so. 
I'lifortunatoly I have hitherto Iieen unable to prove the truth of this except l)y the laboi'ious method 
in the text. I do not therefore know whether the result remains true for higher degrees of approxima¬ 
tion, although I suspect it does so. If it should be true, it would be very easy to compute the integrals 
when higher powers of f are included. 
It may be worth mentioning that the variables are sejmrable in the integrals. Unis, when B/* •f i"” 
denotes any one of the eight forms, 
M (1 
1 
ff+f 
1 (1 - ^ cos 2<f>y (C,-*)-' H> 
.10 
/i-4)8'rD . siiP^djpy (cp)^ 
V A Jo (I -/i cos 2c/>)^ 
'I'he (f> integrals j/zrescnt no difficulty, but with regard to the others we are met by the impossibility of 
expanding in powers of sec- 0 for the lower orders. It would be a great step in the right direction, if it 
could be proved that all the terms which do not involve the factor —J- necessarily vanish, 
2/ -t-1 
4 A 2 
