ARTICLES 415 



By a well-known homogeneous linear transformation ' we 



f 2 = @iX! + fisXz 



can transform (20), omitting the product terms, into 



where X x , X 2 are the roots of (21). 

 From (23) it follows, firstly, that 



dg-2 X3 £ 2 



(22) 



(23) 



(24) 



I. = K^ (25) 



A„ 



and, secondly, that 



ft f - x -f- 1 

 ft § = ^ J 



(26) 



Hence, by addition, 



where it? is the radius vector of the point ^£2 in a diagram 

 in which these two variables are plotted as rectangular co- 

 ordinates. We thus obtain a graphic interpretation of the 

 quadratic form X^i 2 -f- X 2 £|. The topography 2 of the in- 

 tegral curves (25) near the origin is indicated in fig. 1 for 

 X x <o, X 2 <o (stable equilibrium) ; in fig. 2 for X 2 )> o, X 2 > o 

 (unstable equilibrium) ; and in fig. 3 for X x y o, X x <( o (unstable 

 equilibrium). The arrow heads indicate the direction of travel 

 along the integral curves according to (27). 

 2. Case of Complex Roots. — Here we write 



**=*/* + iy ( } 



X 2 = [x, — iv v ' 



1 See, for example, Liebmann, Lehrbuch der Differentialgleichungen, pp. 99, 



131. 



- Compare Liebmann, loc. cit. ; also von Dyk, Akad. Wiss. Munich Sitzungs- 



ber., 1909, Abh. 15 ; Munich Abhandlungen, 1914, vol. xxvi, Abh. 10 ; also Sharpe 

 Ann. 0* Math., ser. 2, vol. ii, 1910, p. 97. 



