493] ON EVOLUTES AND PARALLEL CURVES. 33 
And let the corresponding numbers for the evolute be 
(to } n , ò , k , t , l ; 1) , a ). 
These are most readily obtained, as in Clebsch’s paper, “ Ueber die Singularitäten alge 
braischer Curven,” Creile, t. lxiv. (1864), pp. 98—100, viz. it being assumed that we 
have 
n" = m + n, l' = 0, 
then by reason that the evolute has a (1, 1) correspondence with the original curve,, 
the two curves have the same deficiency, or writing this relation under the form 
m" — 2 ri' + i” — to — 2 n + i, 
' we have m" = 3m +i, = a ; and the Pltickerian relations then give the values of k", 8", r". 
In regard to these equations n" = m + n, i" = 0, I remark that if we have two 
curves of the orders m, m', and on these points P, Q having an (a, a!) correspondence, 
the line PQ envelopes a curve of the class ma' + m'a, and the number of inflexions 
is in general =0. Now in the present case, taking P on the given curve and Q the 
point of intersection with IJ of the normal (or harmonic of the tangent), the orders of 
the curves are (m, 1), and the correspondence is (n, 1) ; whence as stated m" =m+ n, i" = 0. 
The formulas thus are 
m" = a, 
n" = m + n, 
t" = 0, 
k" = — 3 m — 3 n + 3a, 
a" = 3a, 
D"= D, 
in which formulas it is assumed that the curve has no special relations to the points 
/, J ; or, what is the same thing, that the line IJ intersects the curve in m points 
distinct from each other, and from the points /, J. 
It is to be added (see Salmon’s Higher Plane Curves, [Ed. 2], (1852), pp. 109 et seq.) 
that m of the k" cusps arise from the intersections of the curve with IJ, these cusps 
being situate on the line IJ, and each of them the harmonic of one of the intersections 
in question, and the cuspidal tangent being for each of them the line IJ. The inter 
sections of the evolute by the line IJ are these m cusps each 3 times, and besides 
i points arising from the t inflexions of the curve ; viz. at any inflexion the two con 
secutive normals intersect in a point on the line IJ, being in fact the harmonic of 
the intersection of IJ with the tangent at the inflexion. It was in this manner that 
Salmon obtained the number 3m +1 of the points at infinity of the evolute, that is 
the expression m" = 3m +1 (= a) for the order of the evolute. 
The remaining — 4m — 3n + 3a cusps arise from the points on the curve where 
there is a circle of 4-pointic intersection, or contact of the third order, and in this 
c. vili. 5