78 Dr. James Bottomley on a 



On a Method of Integration applicable to certain 

 partial Differential Equations. By James Bottomley, 

 D.Sc, &c. 



{Received December 2jtJi, i8g2.) 



In this paper is proposed a method of integration 

 applicable to certain partial differential equations, which, 

 possibly, has not been previously noticed, and seems to 

 have the merit of simplicity, also the advantage of being 

 applicable to some equations of high order, even where the 

 general theory belonging to equations of that order has not 

 been investigated. 



The method, shortly stated, depends on the fact that 

 when a function of several variables is differentiated with 

 regard to all those variables, the final result is independent 

 of the order in which these operations are performed ; by 

 the addition of these identical results to certain partial 

 differential equations, we obtain equations which admit of 

 being expressed as equations of lower order. As an example 

 of the method, consider the equation — 



d\ _ dh . 



then the subsidiary equation will be 



cVh _ dh 

 dxdy dydx ' 

 by addition we get 



(2) 



cZ% _l_ _dh_ ^d^ dh 



dji? dxdy dy"^ dydx* ' ' • \ / 



but each side of this equation is a complete differential, 

 for we may write it in the form 



d (dz , dz 



±(d^^dz\^d^rdzdz\^ 



dx\dx dy) dy\dy dx) ' ' * ^' 



