Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/229

CHAPTER XVI

ENVELOPES

130. Family of curves. Variable parameter. The equation of a curve generally involves, besides the variables $x$ and $y$, certain constants upon which the size, shape, and position of that particular curve depend. For example, the locus of the equation

(A)	$(x-\alpha )^{2}+y^{2}\ =\ r^{2}$

is a circle whose center lies on the axis of $X$ at a distance of $\alpha$ from the origin, its size depending on the radius $r$. Suppose $\alpha$ to take on a series of values; then we shall have a corresponding series of circles differing in their distances from the origin, as shown in the figure.

Any system of curves formed in this way is called a family of curves, and the quantity $\alpha$, which is constant for any one curve, but changes in passing from one curve to another, is called a variable parameter.

As will appear later on, problems occur which involve two or more parameters. The above series of circles is said to be a family depending on one parameter. To indicate that $\alpha$ enters as a variable parameter it is usual to insert it in the functional symbol, thus:

$f(x,\ y,\ \alpha )=0.$

131. Envelope of a family of curves depending on one parameter. The curves of a family may be tangent to the same curve or groups of curves, as in the above figure. In that case the name envelope of the family is applied to the curve or group of curves. We shall now explain a method for finding the equation of the envelope of a family of curves. Suppose that the curve whose parametric equations are

(A)	$x=\phi (\alpha ),\;y=\psi (\alpha )$

touches (i.e. has a common tangent with) each curve of the family

(B)	$f(x,\ y,\ \alpha )=0,$