Torsion of a Helix on any Cylinder. 61 
Verification. 
These formulae for p and <r may be verified by the same 
method as was used in the verification for helices on 
cylindrical surfaces. 
We have 
This reduces, as above, to 
rfr 2 [tan- a ./'Cr) 2 -l] = rW0 2 . 
••■ ta„**./V)S-l = ^(g)'. 
Hence the p, r equation of the projection of the helix on 
the plane of xy is 
_ f 
tan 2 a. /'(f) 2 — 1 = - 2 
v — p 2 
•'• P 2 ~ r2 tan 2 «./'(r) 2 ' 
dp__ rf'(r)-ry'(r} 
■'■ Pdr "' tan 3 a.yV) 3 
~ r L tan 2 a./V) S J- 
, _ dr _ p tan 2 a. f(ry 
*'* p ~ T dp ~ tan 2 a m f'(r)*>-f'(r) + rf"(r)' 
p' rfjff Vtan^./Q) 2 -! 1^ 
•*' 9 = sm^ ~ ± tan 2 a .f(rf —f (r) + rf(r) * sin a cos a 
Also o- = /o tan a ; and so we find o\ 
These results verify the values of p and a obtained by my 
method. 
Examples. 
1. Helix on a paraboloid of revolution. 
Here z= y„ if I is the semi-latus rectum of the generating 
parabola. . 
Also r increases with <£; and so we take the plus signs in.-— 
