493] 
ON EVOLUTES AND PARALLEL CURVES. 
35 
n —2/i — <7, or n — 2f 2 — g tangents from the point in question to the given curve; the whole 
number is thus (m -/) + 2/j + (n- 2f x -g) or (m-f) + 2/ a + (n- 2f a -g), -m + n-f-g, 
the class of the evolute. The two values of n" give 
2 n" = 2m + (n — 2/i — g) + (n — 2f 2 — g), 
viz. twice the class of the evolute = twice the order of the curve + the number of 
the focal /-tangents + that of the focal /-tangents; but this is not true for all relations 
whatever of the curve to the absolute. 
The tangents from / to the given curve (excluding the line IJ and the tangents 
at /) are n—2f 1 —g tangents; and similarly the tangents from / to the evolute 
(excluding the line IJ and the stationary tangents through I) are the same n — 2f 1 —g 
tangents; say the curve and the evolute have the same n — 2f x —g /-tangents. Similarly 
they have the same n — 2/ 3 — g /-tangents; or together the same 2{n—f x —f 2 — g), 
= 2(n—f—g) focal tangents. And the curve and evolute have the same (n—2f 1 —g)(n—2f 2 —g) 
foci. 
The foregoing specialities f and g refer, g to the ordinary contacts of the line IJ 
with the curve, viz. the curve is supposed to have with the curve at an ordinary or 
non-singular point thereof a contact or 2-pointic intersection, and f that is f x or f 2 , 
to the multiple points having f x or f 2 ordinary branches, none of them touching the 
line IJ. Thus the formulae do not apply to the cases of IJ passing through a node 
or a cusp of the given curve, or touching it at an inflexion; nor to the cases where 
at / or / the curve touches IJ, or where there is at I or / an ordinary double 
point with one of its branches touching IJ, or where there is at / or / a cusp, 
where the cuspidal tangent is or is not IJ. 
It is easy to see that in the case of a multiple point of any kind whether 
situate on IJ or at I or /, each branch of the curve produces its own separate effect 
on the singularities of the evolute: thus if we have on IJ a double point neither 
branch touching IJ, then the separate effect of each branch is nil, therefore the effect 
of the double point is also nil: but if one branch touch the line IJ, then the whole 
effect is the same as if we had this branch only; viz. we have here the case g = 1. 
And so if there is at / or / a double point with one branch touching the line IJ, 
then the effect of this branch is as if we had this branch only (a case not yet 
investigated) but the other branch is the case f= 1. And so if we have at / or / a 
double point with two ordinary branches touching each other (tacnode or, if the two 
branches have a contact higher than the first order, oscnode), then if the branches 
do not touch the line IJ the case is /= 2, but if they do, then the effect is twice 
that of an ordinary branch touching IJ. In support of these conclusions, observe that 
such multiple points, with ordinary branches, present themselves in the case of two 
or more curves which intersect or touch each other in any manner; and that the 
evolute of a system of two or more curves is simply the system of the evolutes of 
the several curves. 
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