THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 229 



In the next case (L.M.S., 25, p. 213), 



H = 5, r = | rw, s (5v) = 00, y = x (1), 



I (v) = f AlL+) 



a 



2 ,v 



cos- 1 



J. _ "1 O ' " JLr 



Sln 



2s* 



> O \ f \ \ / ' 



to = - I + 6x - x", 19' w = X~, P-i-ta= l ~ 3X , (~*^ a- * a = x~> (a), 



10 



and x 1 is the icosahedron irrationality, and KIEPKRT'S _/"' = ./. 

 In ABKF/S form, with y = x, Z has the factor z x, 



2 



and putting z x -, ABEL'S integral 



^dz = 2l(2v), with k- 



V /Z ;j 



Putting ,s + .r = />', 



! /, A - 2 ! (* + I ) v/ { 2^ :! - ( L - *) ^ 2 

 5 2 5 



- 2 si r i (* ~ 1) x/i 2 ^ 3 +_(!- ) <3 ~ 2 ^ - "I 

 5 2 J 



and the degree of the expressions is halved, with great gain of symmetry. 

 The degree is halved with greater ease by putting s = f 1 in I (2'c), and now 



I (2.0 = 2 cos- 1 <* ~ x) v /{2t * + (1 + *) t2 

 5 2^ s 



= 2 sin -i.(* t^A/l 2 *!- (! +^1 

 derivable from the preceding I (w) by writing - for t, and for x. 



