6 . M. Falk, On the Integration of partial 



lu tlie first case we have for n -- 2r — 1 a system of '1v^:'lJ = 



r (2r — 1) equations, containing- 7-, y' , ... f/'"~''. There is, therefore, only 

 one more equation wanted, in order to be able to eliminate the rn=r(2r-l) 

 functions </ , f/', ... , f/'" ". This desired equation will in general be obtai- 

 ned by eliminating the r functions 7^"' between r-\-l of the n-\-l=2r equa- 

 tions, which involve (/"" . We see, that there are several ways to construct 

 this equation, wherefore also we may get more tlian one diiferential equa- 

 tion from the given primitive. The equation required will be of the form 



z G^ '/:,'/,,... 'fr: <i\, 'h, ••• ',',.. .'ir'\ -/r", ... '/r")=o..(/), 



where J is a determinant, which is linear with respect to the differential 

 coefficients of z of the n'"' order. Now eliminating (p , (f', . . . y/"~') be- 

 tween (/) and the before obtained equations , we get an equation of the form 



where z = — — = — . 



Thus we see, that, when the differential equation is of tlie (2r — 1)" 

 order, it also becomes linear with respect to the diiferential coefficients of z 

 of the (2r — ly* order. We may, moreover, get more than one such diff. 

 equ. from a single primitive. 



Secondly suppose n = r . Then we have («+1) equations involving 

 f/j", f/2", ..., If". As these functions occur in a linear form, elimination 

 between them ordinarely leads to a single equation, which is linear with 

 respect to the difterential coefficients of z of the «"' order, that is: involves 

 a linear function A of these quantities. Tliis equation, then, will be of the 

 form (/), where n is to be put for r or vice versa. Now elimination 

 between this equation and the equations involving y , 7', ... , y"'~^' must 

 remove all these functions and give a single diflerential equation of the r'' 

 order, which is linear in the difterential coefficients of c of the ?■"' order. 



There is also another mode of genesis of a partial differential equa- 

 tion from a primitive. Suppose 



~ = f ('*' , y , «^ , a, , . . . a,) (h) , 



rt.2 , ... a,, being r arbitrary constants. Such an equation Lagrange has 



"1 : 



termed a complète primitive. ' Thence we get by differentiation ^ ^j^ ^ 



