363] 
ON THE THEORY OF THE EVOLUTE. 
475 
I consider the particular case where the conic touches the absolute. There is no 
loss of generality in assuming that the contact takes place at the point (y = 0, z = 0), 
the common tangent being therefore z — 0; the conditions for this are a — 0, h = 0, 
and we have thence (7 = 0, Z = 0. Substituting these values, the equation contains the 
factor 0; and, throwing this out, it is 
Z (- H0 3 + (B + 2G) 0 2 ) 
Y( A0 3 - 2 H0- ) 
+ Z( - A0 2 + 3H0 ~(B + 2G)) = 0, 
or, what is the same thing, 
0 3 ( -H X+ AY ) 
+ 0- ((J5 + 2G) X — 2HY - 
+ *( 
+ ( 
AZ) 
3HZ) 
(B + 2 G) Z) = 0, 
where it will be observed that the constant term and the coefficient of 0 have the 
same variable factor Z, where Z = 0 is the equation of the common tangent of the 
conic and the absolute. The evolute is in this case of the class 3. It at once appears 
that the line Z = 0 is a stationary tangent of the evolute, the point of contact (or 
inflexion on the evolute) being given by the equations Z= 0, (B + 2G) X — 2HY = 0. 
The equation of the evolute is found by equating to zero the discriminant of the 
cubic function; the equation so obtained has the factor Z, and throwing this out the 
order is = 3. The evolute is thus a curve of the class 3 and order 3, the reduction 
in the order from 3.2, =6, to 3 being caused by the existence of an inflexion. 
Comparing with the former case, we see that the effect of the contact of the conic 
with the absolute is to give rise to an inflexion of the evolute, and to cause a 
reduction = 1 in the class, and a reduction = 3 in the order. 
I return now to the general case of a curve 
U = y, z) m = 0. 
Using, for greater simplicity, the equation x 2 + y 3 + z 2 = 0 for the absolute, the equation 
of the normal is 
U = 
X, Y , Z 
x , y , 0 
d x U, dyll, d z ll 
= 0; 
we may at once find the class of the evolute; in fact, treating (Z, Y, Z) as the 
coordinates of a given point, the two equations (7=0, 0 = 0 determine the values 
(x, y, z) of the coordinates of a point such that the normal thereof passes through 
60—2