ENVELOPES 
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the equations of two curves, then u + kv = 0 (where ft is a constant) 
represents a curve which passes through all the points of intersec 
tion of the given curves. Applying this principle to the curves (4), 
we see that a third curve through P is given by the equation 
(5) f{x, y, ao + Acc) - f(x, y, a 0 ) = 0. 
The left-hand side has the value Accfjx, y, cc 0 + $ Acc) (Law of 
the Mean, Chap. V, § 2). Hence the coordinates of P satisfy the 
equation 
(6) fa(x, y, Uq -f- 0 Ace) == 0. 
Now let Ace approach 0 as its limit. The point P approaches the 
point of tangency of the first curve (4) with the envelope, and the 
left-hand side of (6) approaches f a (x, y, ce 0 ). Hence the equation 
/«(*» V’ «o)= 0 
represents a second curve passing through the point of tangency of 
the first curve (4) with the envelope. Thus we obtain the 
Theorem. The envelope of the family of curves 
f{x, y, cc)= 0 
is given by the pair of equations 
( 7 ) fix, y, a) = 0, = f a (x, y, a) = 0. 
Example 1. 
we get : 
Applying formulas (7) to the family of circles (1) 
ff = —2(z-«)=0. 
ca 
Equations (7) now tell us that the envelope is given by the pair of 
equations 
{x — cc) 2 + y 2 = 1, x — « = 0. 
These equations are equivalent to the single equation obtained by 
eliminating a : 
y 2 = 1, or y = 1 and y = — 1. 
The analytic result is seen to correspond with the geometric evi 
dence. 
Example 2. To find the envelope of the family of ellipses whose 
axes coincide and whose areas are constant. 
Here,