ON EVOLUTES AND PARALLEL CURVES. 
[493 
circle is = 0: the parallel curve here breaks up into the original curve, together with 
the focal tangents from the points I, J; viz. we have 
to' = 2n + (n — 2/i — g) + (n — 2/. 2 — g), = 2m + 2n — 2f — 2g ; 
and knowing m', n', i we have k. 
The points on IJ of the original curve are I, J counting as f x and f % respectively; 
or together as f points: the points of contact counting as 2g: and besides m-f—2g 
points. As regards the parallel curve, we have the same points on IJ; but here I 
is (n — g) tuple point, having in respect of each branch of the /i-tuple point on the 
original curve a pair of branches touching each other, and in respect of each of the 
tangents from I to the given curve a single branch, together 2f 1 + (n — 2f 1 — g), = n — g 
branches; and thus counting n — g times: similarly J counts n—g times. Hence also 
for the parallel curve // =f 2 ' = n—g. In respect of each of the points g, we have a 
point where there are two branches touching each other and the line IJ; and thus 
counting 4 times, or together as 4g: moreover, on account of the two branches at each 
of these points, g' = 2g. Lastly, each of the m—f—2g points is a node on the parallel 
curve ; and as such counts twice; m = 2 (n — g) + 4>g + 2 (m — f— 2g), — 2m +2n— 2f— 2g 
as above. 
And we have thus the formulae 
m' = 2m + 2n — 2f — 2 g, 
n' = 2 n, 
t = — 6m + 2a, = 2i, 
k = 2a — 6/— Qg, = 6n + 2k. — 6/— 6g, 
f = 2n — 2 g, 
9' = 2g, 
where observe that m is = 2n"; that is, twice the class of the evolute (which relation 
however is not in all cases true for a curve with singularities); and further that n' —f — g' 
is = 0. 
The case of a curve for which n—f—g = 0 is very interesting and remarkable. 
Recurring to the formulae for the evolute, we have here m" = tc, n" = m, i" = n, 
k" = n — 3m + 3k. And for the parallel curve m = 2m, n' = 2n, i = 2i, k = 2k, f = 2f, 
g' = 2g; formulae which lead to the assumption that the parallel curve here breaks 
up into two distinct curves, each such as the given curve. 
Observe further that for a curve possessing the singularities f and g, but where 
n—f—g is not =0; then for the parallel curve we have as above n—f' — g' = 0; 
or the parallel of the parallel curve should, according to the assumption, break up 
into two distinct curves such as the parallel curve; this is of course correct. 
Consider the evolute of the parallel curve: since for the parallel curve n —f — g = 0, 
the formulae for the evolute thereof (viz. those containing the undetermined quantity ©)