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PROFESSOR J. CASEY ON A NEW 
and eliminating <p between these equations, we have the Cartesian equation of the 
parallel curve. 
123. Since the ordinary tangential equation of a curve is the envelope of the 
line x-H'+jU/y + *= 0, the tangential equation of the parallel curve is the envelope of 
Xx -\-py-\- ‘'i^'sA 2 -{- (Jj 2 = 0 . Hence we have the following theorem : — 
If the tangential equation of a curve is 
F(X,^, 0=0, 
the tangential equation of the parallel curve is 
F(a, p, (252) 
124. If the result of the last article he expanded by Taylor’s theorem, it may be 
written in the form 
P+QK/A 2 -y=0; 
or, cleared of radicals, 
P 2 -Q 2 F(X 2 +^ 2 ) (253) 
Hence the class of the parallel curve is twice the class of the original, and is inde- 
pendent of the sign of k. This shows the figures got by taking k plus and minus are 
both included in the equation of the parallel curve. 
125. As in art. 30 we get for the intrinsic equation of the parallel curve* 
^ = 2/'(<p) cos <p +/"(<?) sin <p+k, 
.*. s=f(<p) sin < P+J/ , (<P) cos <p d<p-\-k (254) 
Cor . From the value of ^ we see that the radius of curvature of the parallel curve 
differs from the radius of curvature of the original curve by the quantity k, as is other- 
wise evident. 
Examples. 
(1) Find the intrinsic equation of the parallel to the curve 
the function on the right being the elliptic integral of the first species. 
* The following is an elegant focal property of parallel curves : — Every single focus of the original curve is a 
double focus of the parallel curve. 
Demonstration. Let a tangent from a point I meet the original curve in two consecutive points P, P' ; then 
if P, P' he the centres of two circles, each of which passes through I, the line IP will he a normal to each, and 
therefore a normal to any curve of which these circles are generators. Now let the point I be one of the circular 
points at infinity ; and since the parallel to any curve is the envelope of a circle of constant radius whose centre 
moves along the given curve, the line IP will he a normal to the parallel curve at I, and therefore a tangent 
at I ; hence if two tangents he drawn to the original curve from the circular points at infinity, these tangents 
will touch the parallel curve at the circular points. Hence the theorem is proved. — November 1877. 
