492 
NOTE ON THE PROBLEM OF ENVELOPES. 
[535 
0 = 0, JJ = 0 touch each other, we have a relation in (x', y, z) which is the locus of 
the point of intersection of the curves ©' = 0 belonging to two consecutive points of 
the curve JJ— 0; that is, the equation of the required envelope is obtained as the 
condition that the curves JJ— 0, 0=0 shall touch each other. But when the curves 
touch each other, they have at the point of contact their derived functions proportional, 
•or we have simultaneously 
JJ = 0, 0 = 0, 
c4© + \d x U = 0, 
dy 0 \d y JJ = 0, 
<4© + \d z JJ = 0, 
the same equations as before, since © and 0' denote the same function. 
It is to be added that, when a = m, the equations 
(4© 4- \d x JJ = 0, 
dy 0 4“ \dyJJ — 0, 
4© 4- Xd z JJ= 0, 
are homogeneous in (x, y, z), and we may by the elimination of (x, y, z) from these 
equations obtain an equation Disct. (0 4- \JJ) = 0, say for shortness A = 0, involving X 
and also the coordinates (x\ y', z). Now it is a known theorem that the condition for 
the contact of the two curves JJ = 0, 0=0 can be obtained by expressing that the 
equation A = 0 shall have a pair of equal roots, or, what is the same thing, by equating 
to zero the discriminant of the function A ; this last-mentioned process leads therefore 
to the equation of the envelope of the curve 0' = 0, viz. (a being = m as above) the 
equation of the envelope of the curve 0' = 0, is in fact 
Disct. a Disct. (w) (©4-XtO = 0, 
viz. we first take the discriminant of the function 0 4- X U in regard to the coordinates 
(x, y, z), and then taking the discriminant in regard to X of this discriminant we equate 
it to zero. This is in many cases a more simple process than that of the direct 
elimination of x, y, z, X from the five equations.