THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 247 



so that 



v = CD, + /Ju,,, u = f'ca, -\- o) (3) 



1 J .J' ./ i i i V"/> 



where f, f denotes real fractions. 



To* obtain this resolution of S we find it convenient to put (L.M.S., 27, p. 449), 



x= m3ot - - (1 ~_2m) " 



(a - m) 2 ' a - m 



and with s = 2 , 



TL T, = (* m \ \ 2t~ T - "" h ~~, (5). 



\ a m/ I a m (a m)* 



Denoting the roots of S = 0, irrespective of order of magnitude, by a~, lr, c~, we 

 can put 



(6), 



__ 

 2(a-m)' 



'Ma. (a. 

 L_ 



a+b = 



a-m a-m 



' ' 



(8). 



With % fj.v congruent to a half-period a>, we can take 



o 2 



and this leads to a relation between a and m, by means of which they are expressible 

 theoretically in terms of a single variable. 

 Also, as shown in L.M.S., 27, p. 450, 



S(W)-S(F)= m * a (10), 



v ' a m 



s ( u y- s (3 r) = (i^ 71 )-* (12), 



m 2 Ul - 2m) a mil m)} 2 



v , , -^~~ 

 (1 2w) 2 (a m) 2 



m-ot. \(l -2m) a m 2 (I m)\~ 

 = __ - 2m -- - _- 



a m \D 



