493] 
ON EVOLUTES AND PARALLEL CURVES. 
41 
in the case about to be considered of a parallel curve; the parallel to any given 
curve is in general a curve not breaking up into two distinct curves of the same 
order with such given curve, and when this is so (viz. when the parallel curve does 
not break up) each normal of the parallel curve is a normal at two distinct points 
thereof: the evolute of the parallel curve is thus the evolute of the given curve 
taken twice; and the parallel curve and its evolute have not a (1, 1) but (1, 2) 
correspondence. Hence, (m, n, 8, k, t, i) the unaccented letters referring to the parallel 
curve, or say rather to a curve which has a (1, 2) correspondence with its evolute, 
and, as before, the twice accented letters. to the evolute, it is not true that 
m" — 2 n" + t' = m — 2n + i; it will subsequently appear that the values of ra", n" are 
correct, those of l", k" suffering a modification; viz. the formulae are 
m " — a-Sf-Sg, 
n" = m + n — f — g, 
*" = /+ 9~®> 
k," = — %m — 3 n + 3a — of — og — ©, 
where, for the present, I leave © undetermined. 
Coming now to the parallel curve, let the numbers in regard to it be m', n, 8\ 
k, r, i; ol, D'. Supposing in the first instance that the given curve does not stand 
in any special relation to I, J, the formulae are 
m! = 2 m + 2 n, a = 6?i + 2a, 
n' = 2 n, 2D' = — 4m + 2 + 2a, = — 2 m + 2 n + a. 
l = 2a — 6m, 
k = 2a. 
Considering the parallel curve as the envelope of a circle of constant radius having 
its centre on the given curve, it appears (e.g. by consideration of the case of the 
ellipse) that when the radius of the circle is =0, there is not any depression in the 
order of the parallel curve, but that the parallel curve reduces itself to the given 
curve twice, together with the system of tangents from the points I, J to the given 
curve : the order of the parallel curve is thus m = 2m + 2n. 
To find the class, consider the tangents from a given point to the parallel curve; 
about the point as centre describe a circle, radius Jc; then the tangents in question 
are respectively parallel to, and correspond each to each with, the common tangents 
of the circle and the given curve, and the number of these is = 2n, that is n = 2n. 
Each inflexion of the given curve gives rise to two inflexions of the parallel 
curve; and the inflexions of the parallel curve arise in this way only: that is i - 2i, 
= 2a — 6m. And the Pliickerian relations then give tc — 2a; a value which maj be 
investigated independently. 
Attending now to the singularities f and g; the values of n, i are unaltei ed. 
to obtain m' we as before consider the particular case where the ladius of the \aiiable 
6 
C. VIII.