1870.] Mr. "W. H, L. Russell on Linear Differential Equations. 15 

 higher powers of (%) than here given. Then we shall have 



Ida? J 



ofa? J 



Substituting these values in the differential equation, and equating the co- 

 efficients of the highest powers of (#) to zero, we have 



v 3 P + p 2 y'=Q, 



whence 



y 



"~~T 



and also 



v s a + 3fV/3 + 2juvy' + v 2 /3' + y" v =0; 

 whence substituting for (v) and reducing, we have 



as before. 



The other integrals given in my last paper may be deduced in a similar 

 way. 



This method suggested to me that it was possible to ascertain if any 



p p 



linear differential equation admitted of a solution of the form y= q eq, 



where P, Q, p, q are rational and entire functions of (#). 

 Let, as before, the differential equation be 



(«« + *,*+.. . . . . .+/3 TO o £3r+ • • .=o. 



Then it is easily seen that the factors of q must be divisors of 



CL m x ; 



hence if we have 



a + a,a?+ . . . +a, m x m = (a:— a) r '(x— b) s ' . . ., 



we must have 



y = ? e *+n,H-V»+ ■ ■ • + 7-^-+ , ^ - + . . . + 7-^,+ ... 



H {x—a)r (x—a)'- 1 (x — b) s 



Now this series can evidently be written in the form 



y = J{(a;)e v io+W+ • • • + V* m , 

 where R(V) can be expanded in descending powers of (#). Hence if this 

 value of y be substituted in the proposed differential equation, we may de- 

 termine tj , y v . . . ^ by the same process as before, 



