CHAPTER VIII 
ENVELOPES 
1. Envelope of a Family of Curves. Consider a family of circles, 
of equal radii, whose centres all lie on a right line. The family is 
represented by the equation 
(1) (x - a) 2 + y 2 = 1. 
where the parameter a runs through all values. The lines 
(2) 
y = 1 and y — — 1 
are touched by all the curves of this family. 
Again, let a rod slide with one end on the floor 
and the other touching a vertical wall, the rod always 
remaining in the same vertical plane. It is clear that 
the rod in its successive positions is always tangent 
to a certain curve. This curve, like the lines (2) in 
Fig. 42 the preceding example, is called the envelope of the 
family of curves. 
Turning now to the general case, we see that the family of curves 
(3) f(x,y,a) = 0 
may have one or more curves to which, as a varies, the successive 
members of the family are tangent. When this is so, two curves 
of the family corresponding to values of a differing but slightly 
from each other: 
( 4 ) /(*> V, <*o) = °> /(«, y, ao + Ace) = 0, 
will usually intersect near the points of contact of these curves with 
the envelope, as is illustrated in the above examples. So if we 
determine the limiting position of this point P of intersection of 
the curves (4), we shall obtain a point of the envelope. We will 
first outline the method and show its application, and then come 
back to a study of the details in § 4. 
From analytic geometry * we know that, if u = 0 and v = 0 are 
* Analytic Geometry, p. 165. 
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