^-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 807 



We may note, in passing, that under the circumstances supposed the discriminant of 

 the quadratic function 



is a perfect square (since the roots are — b x \ka 2 and — d jb^. In fact, we have 



(2b l + c o y-Wa 2 d 



= 4(6 1 2-4a 2 cZ ) + 46 1 c + c 2 



= c n 

 by virtue of (12). 



— r 2 



To find the Conditions in the most General Case that the ^-discriminant 



furnish a tac-locus. 



The necessary conditions will be best seen by using Newton's diagram to 

 estimate the relative orders of the terms in the differential equation. If we assume, 

 as will be justified by the result, that it is unnecessary to go beyond the term in 

 p 1 , we may write our equation (with the same origin and axes as heretofore) 



If we remember that p is of the same order as y/x, we may, for the mere purpose 

 of estimating the order of the terms, write the equation in the form 



x\\oi + c o y-^d o x 2 + e xy-\-f y 2 ) + xy{b 1 x + c^j) + a^f = . . (16). 



Arranging the terms in Newton's diagram, we have figure 2. If two integral 

 curves touch each other at the origin, we must have two approximations of the form 

 y = Xx a ; and there must be a group of at least three effective terms : 

 that is, there must be three outlying points in the diagram in a 

 straight line next the origin. As a first condition, therefore, the 

 term in x 3 must disappear, i.e., b = 0. If this condition alone 

 were satisfied, we should fall back on the case last discussed. In 

 fig. 2. " order that the p-discriminant may not touch the two integral curves 



that have a common value of p at the origin, it is further necessary that c = 0. If we 

 retain only the three effective terms, the differential equation may be written 



d x 2 +\xp+a 2 p 2 = (17). 



If now we put y = Xa?°, we see that a — 2, and 



^ +2J x X + 4a; 2 \ 2 = 0. 



There are therefore two first approximations, viz. : — 



2« 2 y={-& 1 ± v /{(V-4Mo)}^ 2 ( 18 )> 



which correspond to two integral curves which touch at the origin. 



