'28 MR JAMES A. MACDONALD ON 



The system of curves 



« ■-■' ■•■-' - 2c(>/ 3 - a 8 ) + (y - .-•)-'( Sf + 4xy + 2a?) = (3) 



has for discriminant A = (y' — x 2 ) 9 



A A = - 4D = -4(y - ») 2 (3y 2 + 4r y + 2.« 2 ) 



A„ = 2B =-4(2/ 3 -^) 

 A„=-4A=-4» 2 . 



Along ?/ = x A A = A u = A D :£0, the corresponding value of c is zero, and the part 

 y — x = of the discriminant is a part of D = and of B = 0. 



Again, the system c 3 y + c 2 (2y + x) + 3cx + 2x = o (4) has for its discriminant 

 A = Sx(x - iyf, 



1=0 

 A B =-18a?(a:-4y) 2 



l s =-6(2x+ I/ )(x-^ y y. 



Along the branch (x — 4y) 8 = of the discriminant all the functions vanish, and as 

 already indicated (a; — iyf is a repeated factor. 



Since A A = for every point of the plane of x y, it is suggested that, regarded as a 

 function of x and y, the expression on the left-hand side of (4) contains a factor inde- 

 pendent of x and y. In fact, the expression is identically equal to (c + 2) (c 2 y + ex + x). 



Along the branch x = of the discriminant all the functions except A E vanish ; 

 the corresponding value of c is zero, and this part of the discriminant is contained in 

 D and in E. 



It should be noted that the ordinary rule for calculating the value of c does not 

 hold in the last mentioned case ; it holds, however, when the new and correct dis- 

 criminant x(x — Ay) is calculated as well as the new values of A A , A B , and A D . 



Nature of the Value of C for an Envelope. 



With regard to the value of c obtained, we have to remark that if it be indepen- 

 dent of x and y in virtue of the relation between the variables along any branch of the 

 discriminant, then that branch must be merely a particular case of the curves given by 

 the c equation, and not an envelope. An envelope may be a particular case of the C 

 equation ; but the corresponding value of c in this case must be a function of the 

 variables, so that it changes its numerical value as we proceed from point to point of 

 the discriminant. 



For example, the curves 



chj + cx + x = (5) 



and 



e'-' sin x — 'ley + sin % = (6) 



have their discriminants respectively x{x — Ay) and y 2 — sm 2 x. 



