AND PARTIAL DIFFERENTIAL RESOLVENTS. 45 



The general solution of (16) is well known to be 



y^ic^+fa-*)*, .... (17) 



where c l} c z are arbitrary constants to be determined. 

 Integrate (15) with respect to x, then 



m= ^_za> £ _ x (l8) 



2 27 v ' 



Hence equation (17) is a root of the cubic 



y' + ay+- - — e"*=o (19) 



The values of c 1} c z are readily determined by substi- 

 tuting the value of y in (17) in (19); they are as follows : — 



c = — i 



2a} 

 c z = — . 

 27 



Substitute these values in (17), then 



"4r-y-& • • • • ( 20) 



is a root of the cubic (19). 



5. The values of ^ e~ x in (19) can be determined by 

 means of a quadratic, so as to satisfy the classical cubic 



y* + ay + b = o (21) 



This equation will coincide with (20) if 



, € x ia> , , 



0—- e~ x (22) 



2 27 v ' 



