168 
are a particular case of a general family of surfaces to which 
we shall in the next place direct our attention. 
193. To find the equation to a surface which envelopes 
a series of surfaces described after a given law. 
Let u=f(x f y, x, a) = 0 be the equation to a surface 
containing, besides other constants, a parameter a; then if 
we give a particular value to a, the form and position of 
the surface in space will be completely determined ; and if 
we give to it all possible consecutive values, we shall obtain 
an infinite number of corresponding surfaces, usually inter 
secting one another two and two. The surface formed by 
these successive intersections, and having with each one of 
the surfaces the infinitely narrow zone in common which is 
contained between the curves in which that individual surface 
is intersected by the preceding and succeeding ones, (and 
which consequently envelopes each one of the first series of 
surfaces, supposing them all to exist together, and touches 
it according to a curve of intersection) has been named by 
Monge the Envelope. Also the curve in which any two 
consecutive surfaces intersect, he has named the Characteristic 
of the Envelope, because it indicates the mode of its genera 
tion ; thus the characteristic of developable surfaces which 
are generated by the intersections of planes, is a straight line; 
and that of surfaces generated by the intersections of spheres, 
a circle. 
To find the equation to the envelope, we observe that if 
after having given the parameter a determined value a in 
u =3 0, we give it a new value a + Sa differing insensibly from 
the former, we obtain the equation to a second surface differing 
in form and position insensibly from the first, and intersecting 
it in a series of points the co-ordinates of which satisfy the 
equations 
du „ d/u 
u = 0, u +—- .()a + —. 
da a a" 
du 
— h &c. = 0 ; 
da 
(So) s 
+ &C. = 0, 
or u - 0,