the parameter being the same in both cases. The slope of (*A*) at any point is

(C) |
(), p. 80D |

and the slope of (*B*) at any point is

(D) |
(), p. 19957a |

Hence if the curves (*A*) and (*B*) are tangent, the slopes (*C*) and (*D*) will be equal (for the same value of ), giving

or

(E) |

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*). If we then substitute the values of and from (*A*) in (*B*), the result will hold true for all values of ; that is,

(F) |

The total derivative of (*F*) with respect to must therefore vanish, and we get

(G) |

where .

(H) |

Therefore the equations of the envelope satisfy the two equations (*B*) and (*H*), namely,

(I) |
and |

that is, the parametric equations of the envelope may be found by solving the two equations (*I*) for and in terms of the parameter .

**General directions for finding the envelope.**

First Step. *Differentiate with respect to the variable parameter, considering all other quantities involved in the given equation as constants.*

Second Step. *Solve the result and the given equation of the family of curves for and in terms of the parameter. These solutions will be the parametric equations of the envelope.*