170 Mr Bateman, The determination of curves 



The direction cosines of the tangent are 



therefore 



sin a sin 0, sin a cos 0, cos a ; 

 de e = sin add. 



Thus a curve for which — H = — is given by 



<r- p~ a- ° J 



cos 2 - sin ( 20 sin 2 - | - %\\ 



x = a sin ad6 



y — a I sin ac£0 



z= —a Jsm 2 ad0 cos (0 cos a), 

 which on integration give 



l sin ( 



20 cos 2 



2 

 cos 2 - cos f 20 sin 2 - j — sin 2 - cos (20 cos 2 ^ 



a; = — sm a 



y = - sm a 



tan 2 - cos [ 20 cos 2 - 

 2 A 2 



cot 2 - cos ( 20 sin 2 - 



cot 2 - sin f 20 sin 2 ^ j — tan 2 - sin f 20 cos 2 - 



z = — a sin a tan a sin (0 cos a) ; 

 when tan 2 - is commensurable this curve is algebraic." 

 Again, for a curve of constant curvature we have 



JU — I Jl/q (JCoq • 



Now ds = sin a. sin (0 cos a) d0 ; 



therefore 



x = j sin (0 cos a.) d0 



cos- ~ sin I 20 sin- - 1 — sm 2 - sin I 20 cos 2 — 



2/ = I sin (0 cos a) cZ0 



z=j— sin a cos (0 cos a) sin (0 cos a) c?0 ; 

 therefore 



cos 2 ■= cos ( 20 sin 2 - j — sin 2 - cos ( 20 cos 2 ^ 



cos 2 -= 



sin 2 ^ 



2% = - — - sin0(l — 2 cos a) + - — - sin0(l+2cosa)~sin0 



1 - 2 cos a v 7 1 + 2 cos a v ' 



