526 



Mr. Russell on Linear Differential Equations. [June 15, 



of land interpreted in connexion with observations for latitude) involves 

 the unknown error of the chronometer, and makes the ship 1' West or 

 East of the true place for every four seconds of time that the chronometer's 

 indication is in advance of or behind correct Greenwich time. Although 

 I believe that every man who uses a chronometer at sea knows this per- 

 fectly well, I shall not omit to state it in the practical directions which I 

 propose to publish, as the Astronomer Royal, Professor Stokes (' Proceed- 

 ings,' April 27, 1871), and Mr. Gordon (writing in the Mercantile and 

 Shipping Gazette ') are of opinion that an explicit warning of the kind 

 might be desirable in connexion with any publication tending to bring 

 Sumner's method into more general use than it has been hitherto. 



X. « On Linear Differential Equations."— No. V. By W. H. 



L, Russell, F.R.3. Received June 15, 1871. 

 Let us now endeavour to ascertain under what circumstance a linear 

 differential equation admits a solution of the form P log e Q, where P and Q 

 are rational functions of (<r). 



If fa+o^-f . . . .)§ + (/3 +^+ . . . 0^S+ - =0, 

 we have, substituting y — V log e Q, 



{(<vK*+ . . • •)^+(A+^+....)^?+. . .}iog e Q.+R=o, 



where R is a rational function of (#). Hence 



d n P d n ~ l Y 

 t« +a^+ . . . •) ^-K/3 +A#+ . . . .)^T+ =0 ? 



or P must be a rational function satisfying the given equation. Having 

 ascertained its value, we have a differential equation of the form 



Divide this equation by L„ and differentiate, and we have an equation of 

 the form 



cP+Mog, Q XM ^log,Q , rf1og«Q 



M ° dtf* 1 ~ 1 1 dx n + Mn dx ~ U ' 



from which we find Q in all possible cases, since ^I^ElS is a rational 



dx 



function of (#). 



It is impossible that a linear differential equation can in general have a 

 solution of the form y==/(log e fl?) ; for in that case we should have 



K+a ,,w+ . . . . . . o^qp^ 



+ ....=o. 



Let 2=log e #, and the equation becomes of the form 



(VK«»+<V*+ • • • .f^ + {h+h ie *+l^+ . . . .)*jj?>> 



+ =0; ' 



