804 



PROFESSOR CHRYSTAL ON THE 



origin; and let the a?-axis be the tangent to the integral curve of (1). Then, corre- 

 sponding to x = 0, y = 0, (1) must have two zero roots, and no more. Hence a = 0, 

 «! = (), ci 2 d£0. In the most general case the values of the other constants, 6 , c , &c, 

 will be unrestricted, and the ^-discriminant locus will not touch the cc-axis. 



If we confine ourselves to points infinitely near the origin, x will be an infinitely 

 small quantity of the first order ; y infinitely small, of second order at least ; and p 

 infinitely small, but not necessarily of the same order as x or y. (It will, in fact, be of 

 the order « - 1, if a be the order of y, i.e., of the same order as y/x.) 



If we neglect in the equation (1) all quantities that are obviously not of the lowest 

 order, we have in the most general case merely 



the integral of which, subject to the condition ?/ = when x = 0, is 



(3); 



AW 

 a J 



y=±s - 



Hence the integral curve of (1) has a cusp at the origin (fig. 1). 



(4). 



Fig. 1. 



It follows that the ^-discriminant locus is in general the locus of cusps on the 

 integral curves of the differential equation. 



On the Nature of the Integral at a Point on the ^-discriminant Locus where the 

 Primitive in question touches that Locus. 



If we take the origin at the point in question and the tangent to the integral curve 

 --axis, the differential equation will take the form 



b o x + c dJ + d o x °' + (h x + C \V)V + a -if- = ° 



(1), 



where all terms have been omitted which will obviously not be ultimately required for 



"in npproximations. 



