satisfying given conditions. 



171 



cos 2 2 sm 2 - 



2u = cos#(l — 2cosa) + - cos#(l + 2cosa) — cos#, 



J 1 — 2 cos a 1 + 2 cos a 



1z = tan a cos 2 (0 cos a) ; 



as before, if tan 2 - is commensurable, this curve is algebraic. 



The following are some special curves : 

 x = sin 0, \ 



y = -cos<f), 4o-2 



r-1' 



# = log tanh ^ , 



a? = sin sinh + cos cosh 0, 



2/ = sin cosh — cos sinh 0, [• p 2 = 4er, 



*=20, 



a? = ^ sinh 20 sin + f cosh 20 cos 0, ' 



y=\ cosh 20 sin — ^ sinh 20 cos 0, ■ p 3 



z = 2 sinh 0, 



