58 Mr. L. V. Meadowcroft on the Curvature and 
Now dO = sin a. dcf>, > 
dt] = cos a d<f>, y 
A, A 
since TZT'=BZB'=the angle between planes at P and Q, 
each containing the generator and 
the tangent line 
= the angle between the tangent planes at 
P and Q, I e. at P and N. 
= dcf). 
1 dd . dd> 
•\ -= -j- as sill a ~- 
p as as 
sin 2 a . , p'd<b 
= ■ , , since as — —. — -. 
p sin a 
p=p' cosec 2 a. 
A 1 dn dd> 
Again . — = — * ' = cos a ~ 
cr tfS as 
cos a sin a 
/>' 
a- 
=p' 
cosec a sec a. 
P = 
V 
cosec 2 a, 1 
a- 
-p' 
cosec a sec a. J 
<j- 
=p 
tan a, as it should do. 
Hence 
Also 
Verification. 
The usual method of procedure in such a case as the above 
is to project the helix onto the plane o£ xy, and find p' 
the radius of curvature of the plane curve. Let y be the 
angle which the osculating plane at P makes with the plane 
of xy, /3 the angle made by the tangent at P with the same 
plane. 
Then, by a general theorem, 
p __ cos <y 
p' ~ co7"/3' 
In our case 7 = - e> — a, ft= - — a, 
Also o- = /o tan a, and 
This verifies the above formula? for p and a. 
p = p' cosec 2 a. 
