DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



_ A + Y + H f + B f - B -M? - *) + F -I./* _ ^ 

 1 p */ <fy p \dx dy! p \tbt dy/ 



: f 



n da; 



o i 7 i A \ i r^ ' T> \ 



1^1 ,~^ " ' /. \ ^7l/ ^., / J-. "1~ ,- ' _7.. " ' _J.. I 



where 



+ c c -- d;l = o 



^ dr J pq dy ~ 



A = Ap c + H^^r + Bq" Gp Fr/ + C ; 



and in each of these ft has the above-mentioned value. 



4. There are two considerations, initially distinct, but found in the course of the 

 argument to be concurrent, which enable us to obtain a certain set of subsidiary 

 equations : they correspond to the two modes of obtaining the subsidiary equations 

 in AMPERE'S method of solving equations of the second order in two independent 

 variables. 



According to the first of them, we note that the new variable u is as yet limited 

 by no conditions ; it has hitherto remained arbitrary. Suppose it chosen so that 

 A = 0, that is, 



+ Hjyq + Eq- - Gp - Ft/ + O = 0. 



Then the term in jSj disappears from the three equations ; and these (after soint- 

 reductions in which A = is used as well as the identical relations affecting the 

 derivatives of a, b. c,f, y, h) take the forms 



dy 



Gf 4-F^O >. 

 dx d 



=0 



dy 



According to the second of the considerations indicated, we assume that the 

 new variable u, which has been adopted, is an argument in an arbitrary function 

 that occurs in the solution. Then fi^ will, through the term dh/du in its value 

 dh/du -r- dz/du, introduce a triple differentiation with regard to u beyond any 

 differentiation that occurs in the integral equations, while no one of the other terms in 

 any of the equations will introduce more than a corresponding double differentiation 

 with regard to u. Assuming the integral to be of such a form that these differen- 

 tiations give rise to derivatives of the arbitrary function, it follows* that ft will 



* Provided always that the number of derivatives of the arbitrary function in question, as occurring 

 VOL. CXCI. A. C 



