493] 
ON EVOLUTES AND PARALLEL CURVES. 
45 
Bat in the remarkable case where the curve and its evolute have a (1, 2) corre 
spondence, then I correct the formulae by adding — © to the expressions for l, k 
respectively. We have for the evolute of the parallel curve 
m'" = 2m", 
n" = 2 n", 
l” — 21" + (2m + 2 n — 4f) — ©, 
k= 2k" + (2m + 2n - 4f) - ©, 
viz. assuming © = 2m + 2n — 4/, this means that the evolute is the evolute of the 
original curve taken twice. 
A very interesting case is when m=n=f: observe that neither m—/ nor n—f 
can be negative, so that the assumed relation m + №- 2/=0 would imply these two 
relations. We have here for the parallel curve m —2m, n = 2n, t =2i, k =2k; the 
parallel curve in fact breaking up into two curves such as the given curve. And in 
this case the formulas for the evolute assume the very simple form m" = k, n" =f 
L "=f, K " = — 2f+ 3k. 
Whatever the original curve may be, we have for the parallel curve m' = n' =f, 
so that the formulae for the evolute of the parallel curve are of the foregoing form 
m"' = k, n"=f', t" =/'-©, k" = - 2f + 3k - ©, which agree with the above values of 
m”, n", t", k". In the particular case m — n—f, we have © = 0, so that the evolute- 
formulae, if originally written down without the terms in ©, would still be m' =2m , 
n'" = 2 n", i" = 2 l", k" = 2k" ; viz. the evolute is here the original evolute taken twice ; 
as already seen, the parallel curve consisted of two curves such as the original curve, 
and each of these has for its evolute the evolute of the original curve.