1882.] an elliptic function solution. 227 



Conversely it may be shown that the equation 



leads to the differential relation 



dx dy dz 



+ ■ 2 "I 2 = V. 



(i-afy (1-yT (i-^ 3 ) : 



Another kind of generalization may be made, by considering 

 the curve 



as a particular case of the curves 



x n + y n =l, 

 which give the differential relation 



dx dit „ 



1 + J - i = 0. 



(1 -a")'" (I -tff'> 



Put !-*-£. 



then (2x n -lf = l-w n , 



or 2.r n = 1-^(1 ~w n ), 



and 2f = 1 + V(l - w n ) ; 



also w = 2" #y. 



_?_ 

 cfo <7y 2 " dw 



Then 



(1-^ (1-^ V(l-« *') 



and if the integrations extend from the point where 



»<\ > 



x = y=2 ", 



When n = 3, 4, or 6, then a; and y can be expressed by elliptic 

 functions. 



For our case of n = 3, put 



/Q 1 - en r 

 v 1 + cn T 



