1889.] A. Mukliopadliyay— J?ZZzph'c Functions and Mean Values. 223 



The required average value of the volume is obtained hj substi- 

 tuting from (18), (19), (20), (21), (23), and (24) in (17), viz., we have 



J-C _7r/8b 2c8\ 



abc ^^~2\3 " sTb) 



- 3^°« Ha-^^) +3T«^^^~'U^) 



2 O sin \(/ tZ i)/ 

 €8 — sin^ 1^ 



_^ c(a^-i^) r i O sin f cos f tZ i/f 



wherein 



(25) 



and n is expressed in terms of elliptic functions by equation (22). 



These two integrals occuring in the expression for M, can be ex- 

 pressed in terms of Jacobi's functions, viz., putting 



we have 



r 



J \/r^2 sin^ ^' 



1^/ = am f 

 = dn i. di 

 sin \li = sn i 



cos i)/ = CJi i 



(l-^-8sin2^)^ f, 



and 



J'il-k^ sin^ ^ = d,fi ^. dt 



which give, from (22), 



n= ~ ,i-{a^~c^)^~ iJrh\a^-c'^)'^ \ sn%di 



= -—i-(a^-c')U74+$l), 



29 



