ON EVOLUTES AND PARALLEL CURVES. 
43 
[493 
493] 
together with 
’ 2 respectively; 
des m - f— 2g 
; but here I 
point on the 
F each of the 
\-g), =n-g 
i. Hence also 
g, we have a 
IJ; and thus 
nches at each 
>n the parallel 
4-2 n — 2f — 2 g 
which relation 
hat n' —f — g' 
id remarkable. 
n" = m, i" = n, 
:' = 2*, /' = 2/ 
e here breaks 
g, but where 
n'-f'-g' = 0; 
ion, break up 
set. 
n'-f'-g' = 0, 
ed quantity ©) 
are m — к, n — m, t — n — ©, к — n' — 3m' 4 3к — ©, or substituting for m', 
their values, and comparing with the formulae in regard to the evolute, we have 
n', К 
m" = 2a - 6/ - 6g, = 2m", 
n'" = 2m 4- 2n -2f— 2g, = 2n", 
t" = 2n - ©, = 2i" + 2 (n -f-g)- ©, 
k" = -6m- 4n + 6a - 12/- \2g - ©, = 2k" + 2(n-f-g) - ©, 
where m", n", i", k" refer to the evolute. Hence by assuming © = 2 (n -f- g), the 
values of m'", n'", c", k" become 2m", 2n", 2i", 2k", viz. the evolute of the parallel 
curve is the evolute of the original curve taken twice. Observe that in the foregoing 
value of ©, the letters n, f g refer not to the parallel curve, the evolute whereof is 
under consideration, but to the curve from which such parallel curve was derived; 
this value © = 2 (n —f— g) is not a value of © applicable to be substituted in the 
evolute-formuke for the case of a curve which has with its evolute a (1, 2) corre 
spondence. 
Instead of the foregoing case of the f viz. /- and ^-singularities, we may, as 
regards the parallel curve, consider the original curve as having any I- and /-singu 
larities whatever. Suppose in this case (excluding always the line IJ and the tangents 
at I or /) the number of tangents from I to the curve is =n— I, and the number 
of tangents from J to the curve is =n — J, then when the radius of the variable 
curve is = 0, the parallel curve becomes the original curve twice together with the 
(n — I) + (w — J), =27i — I — J tangents ; so that the order is m = 2m 4- 2w — I — JQ) ; we 
have, as before, n = 2n and i = 2t, and these values give k , so that the equations are 
m' = 2m + 2n — I — J, 
n = 2 n, 
l = — 6m 4- 2a, = 21, 
k = 2 cl — I — J, = 6n + 2k — 3I — 3/. 
Suppose 2n—I — J=0\ this implies n — I= 0, n — J= 0 since neither n — I nor 
!/ — J can be negative; viz. that there are no /- or /-tangents; and conversely, when 
his is the case 2n — I — J= 0: and we have then m, n, i, k = 2m, 2n, 2o, 2k ; viz. 
t is assumed, as before, that the parallel curve breaks up into two distinct curves 
uch as the original curve; that is, the condition in order that the parallel curve 
hould break up, is that the original curve has no focal tangents. Observe that the 
lumber of foci is =(n — I)(n — J) which is =0 if only n — I— 0 or n — J= 0; but as 
egards real curves /=/, so that the equations n — I= 0 and n — J=0 aie one and 
he same equation, satisfied if (n — I) (n — J) = 0; so that for a real curve without foci 
real or imaginary) the parallel curve will break up. An instance given to me b\ 
)r Salmon is the curve ^ + ^-c S = 0 or + 27c*V = 0, here m = 6, n = 4, 
1 That the order of the evolute is not (in every case of a curve with singularities) one-half this, or 
=m+n-^ (I+J), is at once seen by remarking that there is no reason why I+J sliou e even. 
6—2