30 MR JAMES A. MACDONALD ON 



Therefore, assuming for the present that A A and A„ are both finite both ways, and 

 that the corresponding value of c is variable, we have the following conclusions: — 

 0, and U„ vanish or do not vanish according as A r and A v respectively vanish or 

 do not vanish ; and when A ( . and A y are not both zero, the curve A = touches at every 

 point of its length one of the curves U = 0. If, however, A,, and A^ are zero in 

 consequence of the relation A = 0, both U x and U„ vanish all along A = (or part of it 

 if it be a degenerate curve), that is A = is (provided c be variable) a locus of multiple 

 points on the system U = 0. Now if A,. = A y = along a finite part of A = 0, A 

 must have a repeated factor. Hence, the occurrence of a repeated factor in the 

 discriminant indicates, under the conditions already stated with respect to A A , A B and 

 c, a locus of multiple pyoints. 



Locus of Multiple Points not in general an Envelope. 



It is easy to show that in general the discriminant curve does not touch the U 

 curves at their multiple points ; for if (x, y) be a double point on U = 0, we have 

 (w, and m. 2 being the values of the tangents of the angles which the tangent at this 

 point makes with the x axis) 



m 1 +m z =-2TJ xy jV !ljl 



(10) 



(11) 



(12) 

 ami 



Iff! )//,/", 2AaA;;/+2(Aa)/A;/ V , 



The condition that the discriminant may touch one or both branches of the C curve 

 at the double point is easily found. In fact, if S'" — be a repeated factor of the 

 discriminant, the condition that 8 = touch one branch of U = at the double point is 



2TJ 3 y _U«, S v S x /-^\ 



U,„, U OT/ Sx s,, 



and the condition that it touch both branches is 



(WfR: ^ m» 



the value of c in terms of x and y having been substituted in U. r ., , U,„ and XJ yy after 

 differentiation. The conditions (14) and (15) may of course be expressed in terms of 

 the discriminant and its derived functions only. 



* 7>) _1_ )n — __ V J " 



i,„+ • • • •) 



1 (c"A„,, + c" 'I 



>,„+ j 



„2A A A„, 





" 2A A A W 





2A A A„ 





The values for the discriminant are : — 





TA A 



(1m — \m ' A-m ' — —1 



, + Z(A A ),A, 



yAin — ;//([ -\-ui., — — VA A 



, + 2(A A )A 



