THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 225 



on putting 



8 -s(nv) = t (12), 



and 



N = 4p 3 \t a (3nv) + * (nv)} 



- 4 MQ { + *(>)} U + .s(m>)-f. C p 

 + MQ {(1 + >,) t + (1 + y) * (y) + a?//} 5 



/MQ_ w'(w) 



V p s>nv)-s(w) 



which also satisfies (16). 



Then 



Mz = ! _ *j3m>)_- 



t 



and 



2 v/K - X /MQ S 4H < + ' 



' 



o 



- 



= P-v ,. 



to \fjivir~ i v \vw * ^' y " " / 



VOL. CCIII. A. 2 G 



'(3m 1 ) ,s(m>) -j- ft 



+ MQ j (1 + ;//)- - 8x - 12,s- (nr) { 3 - 2MQx" (/ 4 r) - MQ./- 1 

 which is a perfect square in the form 



= 1 2 i/pt* - \/MQ |rS + x/MQ/V (nr) 1 (14), 



implying that 



, , , , MQ /MQ *"(>,r) 



(M -()+ ^; = V p 7M 



M Q <n 4. \3_ r l".s(n-r^ = M( ^ K'('ill' 4. 4 / A1( ^ ,-/(,) 



requiring the relations 



1 = l 

 2 



_ __ 



= 3- v - s , (w) ~ (3w - v) _ s (w) - | ., ( 3w) _ >s (nv ) 



T (21), 



