THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 231 



Introducing a normalising homogeneity factor M, so that the substitution 



1 z I \ 

 , - , - ) should correspond to (v, 2v, 4?>), 



1 ~~ Z %> i 



M = **(!-*)' (7), 



derived from (L.M.S., 27, p. 453), 



M 3 = MM, M = ? ' s ' < *L- -- = .x -VZL- 8 



1 2!Pv _ 1 6? + 92 3 22 3 2* r>V _ 1 P (r) _ 5 2 z- 



M 3 2 5 (L --s) ; M 3 : M = Uz'(l -- 2 ) ;; 



2v _ I + 62 152 3 + 102 3 2 4 ip'Zv _ P (27') __ 3 9,z + 5z , , 



M 3 z> (l-z) r M 3 " : M : 142(1 :.) 



12^4?: _ - 1 + 62 32 3 22 s 2 4 7>'4-c _ P (4r) _ - I + 3: + :j; 3 



"M 3 2 (1-2)5 M 3 " - 1' M 142*(1-2) S 



JLV"T- ' t~r , - "/ i ( 1 __ K) T fiVll - rr ~ *' ' i 1 " 1 7 



1Q 7 " ^ ' ' 4Q "/ ' 



4r i/ / / 



I 8wi) GOJ _., /n 

 6XP 49 "Y : ' ( 



In the notation of Krj-HN-FiuCKE, ' Modulfunctioneu,' 2, p. 39!), 



Pt? + *2'-h V'4r _ (!, /rn 



M 3 M~ 



T = L~^ i +_5|_+^ ==5 + 2+1 ^ 2 + 2 7 1 (14). 



10. With (L.M.H., 25, p. 232) 



/.-; -=7 o. 



= is a unicursal (/-, in which 



and from the relation 



P(4r)+ P(5-') = (4), 



(5), 



L8P(2 W ) = 4 + 2(1 + y) + 4^ + - - J) (1 + 



= 1 + - 3jr + 7/> 3 



