NOTES ON THE p-DISCRIMINANT OF ORDINARY 

 LINEAR DIFFERENTIAL EQUATIONS 



By Arnold Emch 



1. In a fundamental memoir^*^ Darboux has proved that in 



general the resultant g[x, y)=0 obtained by the elimination of 



dy , 

 p=^ between 

 ^ ax 



and 



<l>{x,y,p)=0 (1) 



^-^=0, (2) 



where ^ is a polynomial in x, y, p, represents in general the locus 

 of the cusps of the integral curves/^) 



To judge from most text-books on differential equations, Dar- 

 boux's important results seem to be unknown to the majority of 

 writers. The chapters on singular solutions are usually based on 

 Cayley's paper : "On the theory of the singular solutions of differ- 

 ential equations of the first order," which first appeared in the 

 Messenger of Mathematics, Vol. II (1873), pp. 6-12. (Also in Col- 

 lected Mathematical Papers, Yol. VIII, pp. 529-534). It will be 

 noticed that Darboux's and Cayley's investigations were published 

 in the same year. Darboux, however, presented his results to the 

 Societe Philomatique in the session of November 23, 1872. Cayley 

 did not discover that the ^^-discriminant, in general, represents the 

 cusp-locus. Picard in Vol. Ill, pp. 44-52, of his Traite d' Analyse 

 gives an elegant analytical proof for the cusp-locus. ^^^ 



(1) Sur Les Solutions Singuliferes Des Equations Aux Deriv6es Ordinaires Du Premier 



Ordre. Bulletin des Sciences Math6matiques et Astronomiques, Vol. IV, pp. 168- 

 176(1873). 



(2) In what follows O is used as a partial differentiation sign. 



(3) See also Dr. Schlesingrer's Einfiihrune in die Theorie der Differentialerleichunsren, pp. 



257-270, (Sammlune Schubert). 



