p-DISCRIMINANT OF A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 805 



The ^-discriminant is given by combining (l) with 



& 1 « + c ]: y + 2« 2 ^=0 (2), 



and regarding p as an arbitrary parameter For all points with which we are concerned 

 the value of p is small. 



Eliminating p, we have, therefore, for an approximation to the p-discriminant near 



the origin 



b o x + c Q y + d o x i ——(b 1 x + c 1 yy = (3). 



Hence the tangent to the p-discriminant at the origin is b Q x + c y = 0. 



The necessary and sufficient condition that the primitive touch the p-discriminant 

 at the origin is therefore 



b = (4). 



The equation to the jp-discriminant now reduces to 



wherein the term d x 2 cannot now be neglected. To the present order of approxima- 

 tion (5) may be written 



J fa*, \ ■ ■ (6)- 



= kx 2 , say. J 



The differential equation reduces to 



c$+d x 2 + b 1 xp+a 2 p 2 = . (7). 



If we assume the approximation 



y = \x a (8) 



for the solution of (7), we get 



a 2 a 2 X 2 ^°" 2 +(a& 1 +c )Xa? l 4-rf a3 2 =0 . . (9), 



where a 2 =£:0, and in the general case cZ 0= jzO. 



Now 2a — 2> = <a, according as a> = <2. Hence, if a<2, we have 2a-2<a<2; 

 and, if «>2, 2a - 2>a>2. Since a 2Z fcO, and d ±0, neither of these hypotheses leads to 

 an approximation. 



We must therefore have a = 2, and 



4a 2 \ 2 +(2b 1 +c )\+d = . • • (10)- 



