272 UNIVERSITY OF COLORADO STUDIES 



S (f> 1 X 



but as -^-7- and h i> — j do not vanish, generally, the condi- 

 tion reduces to 



V^=0. (10) 



We have therefore the theorem, to the simultaneous equations 



correspond dualistically the equations 



But to a point of inflexion corresponds dualistically a cusp, hence 

 Darboux's theorem: 



The equation resulting from the elimination of p between <t>i=0 



and -K— =0, or the jh-discriininunt of the differential equation 



<f>i=0 represents in general the locus of the cusps of the integral 

 curves. 



3. At the end of his paper, referred to above, Cayley states: 

 " By v^hat precedes, it appears that the ^^-discriminant locus is made 

 up of the envelope locus, cuspidal locus, and the tac-locus; as I in- 

 fer, each of them once." (It ought to be twice for the tac-locus). 

 This proposition and a similar one concerning the c-discriminant of 

 a system of curves, f{'.r, y, c) =0, were given without proof by Cay- 

 ley. J. M. Hill proved them in an elaborate article published in 

 the Proceedings of the London Mathematical Society^') of 1889. 



That the tac-locus, if it exists, occurs twice in the7J>-discriminant, 

 was proved by Darboux in his fundamental memoir of 1872. As an 

 example Darboux assumes the system of integral curves of a special 

 differential equation in the form 



X^A+\B+C=rO, (11) 



where A, B, and C are functions of x and y, and \ is the constant of 



(1) On the c- and p-discriminant of Integrable Differential Equations of the First Order, 

 Vol. XIX, pp. 561-589. 



