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IV. — The C Discriminant as an Envelope. By James A. Macdonald, M.A., B.Sc, 



(Read 5th July 1897.) 



The purpose of this paper is to discuss the conditions under which the C dis- 

 criminant of a system of curves furnishes a curve which at every point of its length is 

 touched by a curve of the system. 



Subsidiary Proposition. 



The following proposition will be used : — 

 If A be the discriminant of 



U=Ac" + Bc"- 1 + ~Dc"- 2 + . . . . +Nc 2 + Pc+Q = (1) 



where A, B, etc., are finite, continuous, single valued, differentiable functions of x and y, 

 the doubled root of U = is 



c=A a /Ab=A b /A d =A p /A . (2) 



Now, if we give such values to A, B, etc., that one of the functions A A . . . . A r 

 vanishes in virtue of A = 0, then it follows from (2) that all the functions except A Q 

 must vanish. If A Q does not vanish, the doubled root is c = 0. Hence P and Q must 

 contain a factor in common with A, and this factor is the particular case of the system 

 (1) obtained by putting c = 0. 



If all the functions A A . . . . A Q vanish, then since A r = 2A A A, r and A y = 2A A Aj,, 

 A r = and A y = all along this part of A = 0, that is A = has a repeated factor. 



Similarly, if A A ^0, A B = 0, the rest of the functions must vanish ; the doubled root 

 is c = oo, and A and B have a factor in common with A, this factor being a particular 

 case of system (1). # 



* The discriminant may always be written in either of the forms : — 



A=A X + B\p (a) 



AsPf + Q x ' (b) 



where x, x' are determinants whose first columns contain respectively only the coefficients A, B, D, and N, P, Q ; and 

 i//, if/ are determinants whose first columns contain respectively only A, B and P, Q. 

 Differentiating (a) we obtain : — 



A a =X+Axa+B^ a 



A e = ^ + Axb + B<(/e (c) 



Ad = Axd + Bi{/d 

 etc. 

 It is evident from (a) that A = always passes through the points common to A = 0, B = 0, and that if these co- 

 efficients have a common factor a contains this factor. In this case (c) shows that A B , A D , etc., vanish. 



If A, B, and D have a common factor, x also contains this factor, and the same factor is by (a) repeated in A ; 

 (c) shows that in this case all the derived functions vanish. If one of the functions B, D vanish identically, it may 

 be considered as divisible by the factors contained in A. 



Similar propositions hold with respect to N, P, Q, and in fact all the results obtained above may also be obtained 

 in this way. 



VOL. XXXIX. PART I. (NO. 4). F 



