William Anthony Granville

“ 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
touches (i.e. has a common tangent with) each curve of the family
the parameter being the same in both cases. ”

“ Hence if the curves (A) and (B) are tangent, the slopes (C) and (D) will be equal (for the same value of ) , giving
or
By hypothesis (A) and (B) are tangent for every value of ; hence for all values of the point given by (A) must lie on a curve of the family (B) . ”