1883.] of Elliptic Functions. 13 



and the resultant multiplier 



M=NN' = ^L; 



so that if u = Kv, then -^—rK'v. 



For instance, if j — J 15 = v/o # 3, 



then #(*#).= sin 18°; 



iT /5 



but if ^"A/g, 



then v' (&&') = sin 54°, 



(Joubert, Comptes Rendus, t. 50); 



, 7 r , 7 + 3V5 



and lick = ^ , 



lb 



c=| = (7 + 4V3)(4-yi5), 



V2c=(2 + V3)(V5-V3). 

 Then, if a; = en Aw, 2/ = en § (Z"+ 32iT') v, 



1+v /1+ 



or 



* = V (tC) q . [ 



y 1 + a; ^ \ i x 



ic ic ^ n]—ic' 



where aft = — ic 



are consistent equations; and lead to 



dtf IP fa? 



(i-tf)(i+t) (i_^(i + |y 



where M=%(l+iJlo). 



