190 
CALCULUS 
2. Envelope of Tangents and Normals. Any curve may be re 
garded as the envelope of its tangents. Thus the equation of the 
tangent to the parabola 
(1) 
at the point (x 0 , y 0 ) is 
m / N 
y - Vo = — (a - x 0 ) 
or 
(2) 
Hence the envelope of the lines (2), where y 0 is regarded as a pa 
rameter, must be the parabola (1), and the student can readily 
assure himself that this is the case. 
The evolute of a curve was defined as the locus of the centres of 
curvature, and it was shown that the normal to the curve is tangent 
to the evolute; Introduction to the Calculus, p. 266, § 4. Hence the 
evolute is the envelope of the normals, and thus we have a new 
method for determining the evolute. 
For example, the equation of the normal to the parabola 
y = x 2 
at the point (x 0 , y 0 ) is 
or 
x — x 0 + 2 x 0 (y — 2/ 0 ) = 0 
x + 2 x 0 y — x 0 — 2 = 0, 
and we get at once as the envelope of this family of lines: 
y = 3 Xq + x = 4 x 0 , 
or 
0y - h) 3 = H* 2 - 
The result agrees with that obtained, l.c., p. 264. 
EXERCISES 
1. Obtain the equation of the evolute of the ellipse: 
x = a cos (j>, y = b sin <f), 
as the envelope of its normals. 
2. The same question for the hyperbola 
x = a sec <f>, y = b tan </>. 
3. Obtain the evolute of the cycloid : 
x = a (6 — sin 6), y = a (1 — cos 6).