34 
ON EYOLUTES AND PARALLEL CURVES. 
[493 
manner the number of them was found, Salmon, [Ed. 2], p. 113, in the particular case of 
a curve without nodes or cusps, and generally in Zeuthen’s Nyt Bidrag &c., p. 91; the 
number of the points in question, in the foregoing form — 4>m — 3n + 3a, is also obtained 
in my Memoir, “ On the curves which satisfy given conditions,” Phil. Trans. (1868), 
pp. 75—143, see p. 97, [406]. 
It is further to be noticed that the m + n tangents to the evolute from either of 
the points I, J are made up of the line IJ counting to times (in respect that it is 
a tangent at each of the above-mentioned to cusps) and of the n tangents from the 
points in question to the original curve. Or taking the two points I, J conjointly, say 
the 2to + 2n common tangents of the absolute and the evolute are made up of the 
line IJ (or axis of the absolute) counting to times, and of the 2n focal tangents of 
the original curve. The focal tangents of the original curve and of the evolute are 
thus the same 2n lines; and the two curves have the same foci. 
The above are the ordinary values of to", n", l", k!' , but if the given curve touch 
the line IJ, then the evolute has at the point of contact an inflexion, the stationary 
tangent being the line IJ; and if the given curve pass through one or other of the 
points I, J, the evolute has in this case an inflexion on the tangent at the point in 
question, this tangent being the stationary tangent of the evolute: but observe that 
the inflexion is not at the point I or J in question: and for each inflexion there is 
a diminution = 1 in the class, 3 in the order, and 5 in the number of cusps. Suppose 
that the point 7 is a / 2 -tuple point on the given curve; then the evolute has / 
inflexions; and similarly if the point 7 is a / 2 -tuple point on the given curve, then 
the evolute has f 2 inflexions. Hence writing f 1 +f 2 = / we have thus / inflexions; and 
if moreover the number of contacts with the line IJ be = g, then we have on this 
account g inflexions; or in all f+g inflexions, and the formuke become 
to" = a - 3/- 3g, 
n” = m + n — f — g > 
= /+ 9* 
k," = — 3to — 3 n + 3a — of— 5 g. 
It is to be noticed here that the number of the intersections of the given curve with 
the line IJ (other than the points 7, J and the points of contact) is = to-/ 1 -/ 2 - 2g, 
that is m—f—2g: each of these gives as before a cusp on the evolute, the cuspidal 
tangent being 77; we have besides on the line 77 (in respect of the g contacts) 
g inflexions, the stationary tangent being the line 77; and each of the i inflexions 
gives for the evolute a point on the line 77; hence the whole number of intersections 
with the line 77 is 3 (to-/-2g) + Sg + ¿, = Sm + i-Sf-Sg, which is thus the order 
of the evolute. 
The tangents from the point 7 or 7 to the evolute are the line 77 counting 
to — /— 2g times in respect of the cusps on this line and 2g times in respect to the 
inflexions, that is to —/ times; the tangents at the point in question to the given 
curve each twice as touching the evolute at an inflexion, 2f or 2/ 2 : and the remaining