56 Mr. L. V. Meadowcroft on the Curvature and 
These two definitions may be proved to be equivalent by 
means of the following well-known theorems : — 
Theorem 1. 
If the tangent to a curve makes a constant angle a with a 
fixed straight line in space, then cr = p tan a. Also the 
binormal makes the constant angle -^ — a with the same 
straight line. 
This is best proved by elementary geometry. 
Theorem 2. 
If the ratio p/a be constant, the tangent-line and binormal 
make constant angles with a fixed direction. 
I shall give a proof of this theorem by means of Serret's 
Formulae. Let (Z, m, n) and (X, //«, v) be the direction cosines 
of the tangent and binormal at any point, Then by Serret's 
formulse we have 
dl d\ _ dm da » dn dv 
.'. dividing each by a and integrating, we get 
t-l + \ = constant = A, 
m + fi = constant = B, Y 
•—n + v = con stant = C . 
a 
Multiply by X, /jl, v and add : .'. AX + B/a + Cv=1.-> 
Multiply by I, m, n and add : .*. AZ + B>?i + Cn = P - ■ 
.*. the tangent line and binormal make fixed angles with 
ABO 
the fixed direction — -=======, which proves the theorem. 
VA 2 -|-B 2 + C 2 
Having proved the equivalence of the two definitions by 
means of these two theorems, it is permissable to use the 
properties which follow from both in the following work. 
I take the fixed straight line as axis of z. Let a be the 
constant angle made by the tangent line with the axis of z. 
Then ~ = cos a = constant. .*. ~ = 0. .*. the principal 
normal is perpendicular to the axis of z, and so parallel to the 
plane of xy. Now the tangent line is perpendicular to the 
