363] 
473 
363. 
ON THE THEORY OF THE EVOLUTE. 
[From the Philosophical Magazine, vol. xxix. (1865), pp. 344—350.] 
According to the generalized notion of geometrical magnitude, two lines are said 
to be at right angles to each other when they are harmonics in regard to a certain 
conic called the Absolute ; this being so, the normal at any point of a curve is the 
line at right angles to the tangent, and the Evolute is the envelope of the normals. 
Let the equation of the absolute be 
© = {a, h, c, f g, Ii#oc, y, zf = 0, 
and suppose, as usual, that the inverse coefficients are {A, B, G, F, G, H). Consider 
a given curve TJ={ y, z) m = 0, and suppose, for shortness, that the first differential 
coefficients of U are denoted by L, M, N. Then we have to find the equation of 
the normal at the point (x, y, z) of the curve U=0. 
The condition that any two lines are harmonics in regard to the absolute, is 
equivalent to this, viz. each line passes through the pole of the other line in regard 
to the absolute. Hence the normal at the point (x, y, z) is the line joining this 
point with the pole of the tangent. Now, taking (X, Y, Z) as current coordinates, the 
equation of the tangent is 
LX + MY+NZ= 0, 
the coordinates of the pole of the tangent are therefore 
(A, H, G^L, M, N) : (H, B, FJL, M, N) : (G, F, G^L, M, N), 
and the equation of the normal is 
Z F Z 
X y Z 
(A, H, G\L, M, iY), (H, B, F\L, M, N), (G, F, G\L, M, N) 
= 0. 
c, v. 
60