1871.] Mr. Russell on Linear Differential Equations. 527 

 and putting for z successively z-\- 2iri, z + Ani, . . . the equation becomes 

 (a +a 1 e*+a 2 e**+ . . . .) + 2ff ( ) + (b + b x e> + bjr* + .... 



dz^ + ' ' * ' - ' 



(a + a 1 e* + a 2 e-+ . . . .) d * f{z +^ + (b + \e* + + . . . .) 



0, 



&c. = &c, 



where these equations can be indefinitely continued. 



Let us now see what are the conditions that a linear differential equation 

 can admit of a solution y = V-\-\/ Q, where P and Q are rational functions 

 of (a?). It is evident that P and Q must satisfy the differential equation 

 separately, so that we may confine ourselves to the case of y=^J/Q. 



We observe that the factors of Q must also be factors of the coefficient 

 of the highest differential ; i. e. if 



a + a^ + a 2 cc 2 + =(#— a) w, (a?— b)* ^=(^-^(^-5)" 



Let x—a-\-z in the differential equation, and expand y in ascending 

 powers of (z). We have then an equation to determine ft, and v . . . . may 

 be found in the same way. Let 



(as + 2) O 2 - 1 + (V + 2x- + 2cc -2)^-#y=0. 



CLOG Q.X 



Let £=#+2, and the equation becomes 



^_4~ + 3)^ + (^ 3 -4/- + 6^~f))^-(^-2)y=0. 



Lety=As M + B.?" +1 + • • • v\ hich gives 3n(n— l)—6n = 0, whence n=0, 

 or 3. 



Let ce—z+ 1 : 



^ + 2)(^ + 3)^ + ( 2 3 + 5^ + 9^+3)^-( 2 +l)y=0. 



Here, putting y as before, we have 6n(n— 1 ) + 3?z = 0, whence ?i — 0, or J. 

 Lastlv, let x = z— I : 



dz~ dz 

 Here 2w(w~l) + 3w = 0, or m = or — Hence the possible forms are 

 / 1 V.^T* 



<jr+l)(#-2)^f+(i»--f+*-3)2-(*-i)y*o. 



the last of which succeeds. 



(.r-f2) 3 (a?+2) 3 Va;-l 



* And also V^-l, K -^L, K ^ I \ — W. II. L. R. — June 30. 



