THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 227 



so that ABEL'S recurring formula for </, (' CEuvres,' 2, p. 157), 



s . + s ^,le?, + -9B (35) , 



becomes 



' " 



(36), 



p"v<> 



'/'"-! 5" '/'-! 



and on comparison with the Weierstrassian formula, 



P (m + 1 ) n> + P (m 1) 10 2P?0 = , pW|P , + - w (37), 



3 



we infer that 



p?7 (38). 



From another formula, 



(39), 

 w) a 



ABEL'S relation 



c*_j _ </,</,_! _ {S J ('//i + 1) zv P/y} (tfmw Pw) 

 p,,,-\ p i^'w 



(40)j 



2 tp'to Pm; P//' 



and 



, P'HIZO pV ..,, - Y -r 



-A-? =: iL(m -\- 1 ) ir limn- ILW 



2 



= P (wt + 1 ) r - Pm/r - P/r (41) ; 



so that 



(;_! = a - C '"-J = 1 a - P (m -f 1) w + P (WJ.M-) + P () (42). 



j'' - 1 



Writing /A for ABEL'S n -\- 2, his k is given hy (2, p. 161) 



M* = G 1) (jfo + ft + + ^-a) 



= (/A 1) (/A 2) Ja + P (/u, 1) w (p. 1 ) Vw 



= l M a-/xP(w) (43), 



as before in (33), since 



(44). 

 With 



x y y 



k = - 2P (w) + * (46). 





 2 G 2 



