p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 813 



the original family and of the orthogonal trajectories. The condition for the existence 

 of a tac-locus where there is no contact with the ^-discriminant locus is that the 

 ■^-discriminant locus, the locus of inflexions, and the locus of inflexions on the orthogonal 

 trajectories must have a branch in common* 



Discussion of the Equation 

 c y + d x 2 + b x xp + aflf- = . 



Since this equation gives the first approximation in the general case to the form of 

 an integral curve in the neighbourhood of an envelope, it is a matter of some interest to 

 investigate the general nature of its integral. 



Since a 2 4=0, we may put a 2 = 1. Dropping the suffixes we may write 



ay+bx 2 +cxp+p 2 =0 (29). 



When the Equation (15) has an Envelojie Singular Solution its Integral is a Family 



oj Algebraic Curves (Parabola). 



If we put £ = .x 2 , vs=zdyjd^, (29) becomes 



ay + (b + 2cm + 4w' i )£=0. 



If we solve for vs and put y = v%, we get 



jfo _ — c± ^/(c 2 — 4& — 4tav) 

 *di~~ ~~4 : 



whence 



4adv _i_^_n 



c+4v± J(<?-4b-4av) + J 



Let u— ± y/(c. — 46 - iav) : then we have 



— 2udu ,d£_ c . 



ac + c 2 —4b—u 2 — au £ ~ 



* The general existence of the cusp-locus of the integral family of a differential equation of the first order was 

 indicated as early as 1851 by De Morgan (Camb. Phil. Trans., vol. ix. pt. 2, p. 113). The earliest absolutely explicit 

 statement of the theorem seems to have been made by Darboux (Comptes Bendus, t. lxx. p. 1331 ; also t. lxxi. 

 p. 267, 1870). In an extremely interesting paper in the Bulletin des Sciences Mathematiques, &c, t. iv., 1873, p. 158, 

 Darboux establishes most of the propositions above given. It is surprising that Darboux's work does not seem to 

 have attracted the notice of Catlet. Reference may also be made to Clebsch, Mathematische Annalen, Bd. vi. p. 211, 

 1873 ; and to Clebsch's theory of " Connexes," Vorlesungen uber Geometrie, Bd. i. p. 1014 et seqq. We have thought 

 it worth while to deduce these results throughout by the approximative method first employed by Briot and 

 Bouquet, because this method is a general one, applicable to the discrimination of special cases, such as arise when an 

 envelope is also a cusp-locus or a tac-locus, &c. ; and because this method is little used by English mathematicians. 



