474
ON THE THEORY OF THE EVOLUTE.
[363
The formula in this form will be convenient in the sequel; but there is no real loss
of generality in taking the equation of the absolute to be x 2 + if- + z 2 = 0; the values
of {A, B, G, F, G, H) are then (1, 1, 1, 0, 0, 0), and the formula becomes
X, 7, Z =0;
X, y, z
L, M, N
where it will be remembered that (L, M, N) denote the derived functions (d x U, d v TJ, d z U).
The evolute is therefore the envelope of the line represented by the foregoing
equation, say the equation fl = 0, considering therein (x, y, z) as variable parameters
connected by the equation U — 0.
As an example, let it be required to find the evolute of a conic; since the axes
are arbitrary, we may without loss of generality assume that the equation of the
conic is xz-y 2 = 0. The values of (A, M, N) here are (z, —2y, x). Moreover the
equation is satisfied by writing therein x : y : z — 1 : 6 : 6' 2 ; the values of (A, M, N)
then become (6 2 , — 26, 1) and the equation is
= 0;
6
Y
1
X
or, developing, this is
H6+ + 2B6 3 - F6 2
+ Yf A6 2 -2H6 3 + G6 2
H&-2B6 + F'
- A6 3 + 2H6' 2 - G6
= 0,
which I leave in this form in order to show the origin of the different terms, and in
particular in order to exhibit the destruction of the term 6 2 in the coefficient of Y.
But the equation is, it will be observed, a quartic equation in 6, with coefficients
which are linear functions of the current coordinates (X, F, Z).
The equation shows at once that the evolute is of the class 4 ; in fact treating
the coordinates (X, Y, Z) as given quantities, we have for the determination of 6 an
equation of the order 4, that is, the number of normals through a given point (X, Y, Z),
or, what is the same thing, the class of the evolute, is = 4.
The equation of the evolute is obtained by equating to zero the discriminant of
the foregoing quartic function of 6; the order of the evolute is thus = 6. There are
no inflexions, and the diminution of the order from 4.3, = 12, to 6 is caused by
three double tangents.
