361 Dr. C. J. Hargreave on Differential 



I have subjoined to this paper a note containing an independent 

 discussion of this somewhat remarkable partial equation. 



10. There are, it is obvious, some cases in which the partial 

 equation admits of a particular solution being easily found, so 

 as to be available for the solution of the ordinary differential 

 equation ; and it may also in many cases be solved under some 

 restrictive hypothesis, the result indicating whether the hypo- 

 thesis is a legitimate one. In fact, it follows the analogy of the 

 partial equation of the first order which we make use of in the 

 integration of equations of the first degree. (See Boole's Diff. 

 Eq. chap, v.) 



11. Before proceeding to apply these formulae to the solution 

 of examples, I think it useful to call attention to the result of 

 eliminating R (as it appears explicitly) from the two equations 

 of the fundamental system. If we make 



II= — -*-— > M=^-s-^J 



dx dy dx ' dy * 



the elimination of R gives 



Mn+^(M+n)+0 2 =o, or (M+<h)(n+&)=R 2 . 



If therefore we know in what manner x and y enter into -ar, we 

 can ascertain in what manner they enter into /x, without inqui- 

 ring what functions of these forms or and p respectively are. 



y y 



Thus suppose we have </>! = #, 2 = ^g, and therefore R 2 = a 2 — ^; 



x x 



y 

 if we know that II is — — in consequence of 37 being a function 



of -, we infer 

 x 



function of xy. 



y y 



of -, we infer at once that M is -, and therefore that a is a 

 x x 



Modes of Solution, and Examples* 

 12. The partial equation expressed in the usual notation is 



and 



There are two cases in which a particular solution of this par- 

 tial equation presents itself spontaneously. First, if P be a 

 function of x only (which call ijrx), the solution of 



dx^++ X dx- = ° 



