THE C DISCRIMINANT AS AN ENVELOPE. 29 



Both these when equated to zero give constant values for c, and therefore do not 

 supply envelopes. 



Envelope and Particular Case of the ' C Curves simultaneously. 



The system (5) consists of straight lines passing through the origin, and two 

 constant values of c correspond to any one of these straight lines ; but along x — Ay = 

 these two values become equal, hence this locus is included in the discriminant. The 

 curves given by the C equation might all touch each other at a fixed point, and in this 

 case the discriminant if it corresponded to a constant value of c would also touch each 

 curve of the group ; but it is plain that it is no more an envelope than the line 

 x — Ay = is an envelope of the group (5). 



On the other hand, the system y = c{x — cf (7) whose discriminant is 3y(Ax :i — 27y) 

 gives for y = 0, c = x. Now, y = is a particular case of the curves indicated by 

 y = c(x-c) 2 , viz., that given by c = 0; but c = is not the corresponding value of c, 

 and the curve y = is a true envelope. In fact we may regard the curve y — as made 

 up of infnitesimally small pieces of each of the curves y = c(x — cf at their maximum 

 or minimum points. 



Main Proposition. 



Let us now take the irreducible equation 



Tj=Ac' + Bc"- 1 + Dc"- 2 + . . . . +Pc + Q = (8) 



A, B, D, etc., being subject to the restrictions already stated 



U, = c"A, + c"- 1 B,+ .... + C P, + Q,. 

 But 



c=A A /A B =A B /A D = etc., 

 .-. ( A B ) 2 = A A A D , ( A B ) 3 = (A A ) 2 A E etc., 



all the functions being assumed finite both ways. 

 Hence 



/A A \" , t,/Aa n "" 



(A A )" 



V *= A {W +B * { ^ ] + 



Similarly 



i.e. along A = 



:g^-{ A.,A,+ A»B,+ +A 9 Q«} 



U » = [f"*y- 1 { A.A+A..B, .... +A.Q,} 

 " (A,)" A "J 



(9) 



