ON THE THEORY OF THE EVOLUTE. 
478 
[36 3 
may write i = 6 +1", where, in the absence of special circumstances giving rise to any 
more inflexions, i" = 0. The system thus becomes 
n — to 2 — 28— 3 k — 9, 
m = 3m ( m — 1) — 68 — 8k —3i 
i 
k = 3m (2m — 3) — 128 — 1 ok — 59 — 21", 
so that each contact with the absolute diminishes the class by 1, the order by 3, and 
the number of cusps by 5. 
I remark that when the absolute becomes a pair of points, a contact of the given 
curve m means one of two things: either the curve touches the line through the 
two points, or else it passes through one of the two points: the effect of a contact 
of either kind is as above stated. Suppose that the two points are the circular 
points at infinity, and let m = 2, the evolute in question is then the evolute of a conic, 
in the ordinary sense of the word evolute. We have, in general, class = 4, order = 6; 
but if the conic touches the line infinity (that is, in the case of the parabola), the 
reductions are 1 and 3, and we have class = 3, order = 3, which is right. If the 
conic passes through one of the circular points of infinity, then in like manner the 
reductions are 1 and 3 ; and therefore if the conic passes through each of the circular 
points at infinity (that is, in the case of a circle), the reductions are 2 and 6, and 
we have class = 2, order = 0, which is also right; for the evolute is in this case the 
centre, regarded as a pair of coincident points. That this is so, or that the class is 
to be taken to be (not = 1 but) = 2, appears by the consideration that the number 
of normals to the circle from a given point is in fact = 2, the two normals being, 
however, coincident in position. 
To complete the theory in the general case where the absolute is a proper conic, 
I remark that, besides the inflexions which arise .from contacts of the given curve 
with the absolute, there will be an inflexion, first, for each stationary tangent of the 
given curve which is also a tangent of the absolute; secondly, for each cusp of the 
given curve situate on the absolute. Hence, if the number of such stationary tangents 
be = \ and the number of such cusps be = /jl, we may write t" = A, + /¿, and there 
fore also l — 9 + A, + fx. 
I remark also that we have 
and therefore also 
— 28 — 3/e = — m (m — 1) + n, 
— 68 — 8 k = — 3m (m — 2) + i, 
—128 — 15/e = — 6m 2 + 15m — 3 n + 3 i. 
The general formulae thus become 
n = m + n — 9 
m' = 3m 4- i — 29 — i 
k = 6m — 3 n + 3t — 30 — 2 l.