H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson :^6B 
Let as usual 
F 
and 
Then we have 
But 
where 
E{k, ^) = I Vl - sin^ 4>dc/>. 
dF 
Mi' = (l-P') 
(-3 
k' = Vi - 
El and F^ denoting the complete elliptic integrals*. 
Thus finally 
f = -^{E,-{l-p^)F,} (Ixxvi), 
and r is known, as soon as p is given, from tables of the complete elliptic integrals. 
Returning to Equation (xx) and putting « = 3 we have 
1 / p2 p* p8 
1 + 
1 + 
and further 
1 1-^2 
2 
1 1-p^ 
2"^~ 
(pY - \ (pY ~ ! ip' 
loge (1 -P^) (Ixxvii), 
= /x„ 
1 — 
1 1 - 
2 p2 
logp (1 — p^) (Ixxviii) 
We now turn to the third moment and may anticipate a recurrence of the 
elhptic integrals. We shall obtain our result on the whole most briefly by 
appealing to Equations (xxii) and (xxiii) on pp. 335 and 336. We have 
p (l-p' ) 3 ,\ 1 /3 3 \| 
=Xi-X3= 2~ p-^U' 2' ^' P )^3^'-^[2' 2' ^' P )\' 
or since 
J 7T J 3 77 
2 2 
p (1 - p^)tt 
.(|,|,2,.^)-l.(|,|,3,,^)j 
2 2 ( 
p (1 -/32) 77 
2 2 
5 
(3 .5 ■■■2^+ 1)2 ^, 
22^- (s) ! (s + 1) ! 
(3. 5. ..2s +1)2 
(3 . 5 ... 2s + 1)2 
22» (s) ! (s + 2) r ' 
22« (s) ! (s + 1) 
* Cayley, Elementarij Treatise on Elliptic Functions, p. 48. 
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