62 Curvature and Torsion of a Helix on any Cylinder, 
the expressions for p and o\ 
r.;pyw.. r i x 
V^ 2 -?cot 2 a 
sir« 
tana .__ r + - 
.*. jo 2 sin 4 a = o- 2 sin 2 a cos 2 a = r 2 — l 2 cot 2 a. 
This is a known result. 
2. Helix on a sphere. 
Here z = v' a z — r 2 . 
If we take z to be positive -jr * s negative. 
r 2 / r 2 
-r.j 2 Y/tan 2 «.- 2 -l x 
.2 • 
r 6 r ra z * sin a. cos a 
■tan 2 a . -H 
z z z 
z-l ; — y on reduction. 
sin a 
\/V — a 2 cos 2 a 
sin« 
n/ r 2 — a 2 cos 2 a 
Similarlv <r _ — 
This again is a known result. 
3. Helix on a cone of semivertical angle j3. 
Here z — r cot /5. 
.-. /(r) = cot/3, /» = 0. 
Also — is positive if the vertex be pointing downwards. 
d<f> 
r cot 2 /3 ^tan 2 «cot 2 /3-l 
* ? — tan 2 a. cot 3 /3 — cot /3 'sin a cos a 
I = (tan 2 acot 2 /3-l) gin2 % ^ ^ _ 1 (tan2 a _ t{m2 ^ sin 2 a cos 2 a- 
p 2- r 2 cot 3 £ 
