44 
ON EVOLUTES AND PARALLEL CURVES. 
[493 
g = 4, k — 6, i = 0, t = 3 : the points I, J are each of them a cusp, the tangents being 
the line IJ ; the number of tangents from a cusp is v — 3, = 1, but for the cusp 
/ or J, this tangent is the line IJ itself, so that we have I = J = 4. 
Theory when the Absolute is a conic. 
When the Absolute is a conic the formulae for the evolute are essentially the 
same as those in the former case, but the formulae for the parallel curve are modified 
essentially and in a very remarkable manner. I observe that corresponding to a 
passage of the given curve through I or J we have a contact with the Absolute, so 
that in the present case f will properly denote the number of contacts of the given 
curve with the Absolute, and attending to this singularity only, viz. considering a given 
curve (m, n, 8, k, l, t; a, D) having f contacts with the Absolute, the formulae for 
the evolute are 
m 
rr 
n 
// 
// 
tc 
In the case /=0, these at once follow from the two equations n" = m + n, and i" = 0. 
The normal is the line joining a point of the given curve with the pole of the 
tangent; or, what is the same thing, it is the line joining the point of the given 
curve with the corresponding point of the reciprocal curve: the degrees of the two 
curves are m, n, and the correspondence is a (1, 1) correspondence. Hence, by the 
general theorem previously referred to, it follows that we have n" = m + n, and l" = 0. 
Compare herewith the demonstration of the theorem in the case where the Absolute 
is a point-pair. 
The formulae for the parallel curve are 
m' = 2m + 2 n — 2f 
n = 2m + 2n — 2f, 
l = 2a — 6/, 
k' = 2a — 6/, 
f = 2m +2 n — 2f 
(so that m' = n\ t = k). The intersections of the curve and Absolute are in this case 
the points f each twice, and besides 2m — 2f points; similarly the common tangents 
are the tangents at f each twice and besides 2n — 2f tangents. Now I remark that 
the parallel curve, when the radius of the variable circle is = 0, reduces itself to the 
original curve twice, together with the 2n — 2f common tangents, and the 2m — 2f 
common points; the order is thus = 2m 4- (2n — 2f), and the class =2n + (2m — 2f): 
and these are the values in the general case where the radius of the variable circle 
is not =0.