THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 257 



so that 



rfw _ n8 /A D A o B - D ,. , C - D , , 



p dt ' ' k \ BC CA AB 



n 8 A 3 * 2 + By + C"z a - D :i S 2 , . 



~k ' ABC 



and this reduces to 



-<fo_n8 j , (A-D)(B-D)(C-I)) S 4 

 p dt k \ p ABC J 



so that 



S ? , + (A-D)(B-D)(C-D), ; 



ABC 



Putting 



2 ' "O Q 



p cr x p,~ p x ,s', t 1\ D 



A- " M- I 2 "M 2 " jyi ~ = M" 



and p = 0, with ,s = cr, 



-_ 

 k \0udr 



4- i (Q^"- 1 + + Q) v 



VOL. CCIII. A. 2 L 



RO= w*v 4 V V AB(! j - --M 



so that 



(A - B) (B - D) (C - D) ,, _ iP / 



ABC; M :i 



and from (G) 1 



TTT = Tit \ 



k J s cr x V /O 



- (! - M ) " ( - ' (r) 



or 



pt v=*I (r) (2S), 



where 



1> S Pr (>')) 



also 



nt = f ]" ^ (30). 



Then 



f '" l( ' >= ^l1 < 31 )' 



(w + v) 



and 



M 2 ^-, = <r f< = F' 9u 



_ o- ( + r) o- (n. - r) _ ff-00 (n + '') ( - '') 



where A. is a homogeneity factor, X = ni/n ; so that as a quasi-Lame function 



