§ 1' 



Genesis of the differential equation from its pritnitive. 



We will examine the differential equation arising from 



F (<p,, Ç,, .. Ç,., x,y, z) = Q (1), 



where i^ is a definite function and (pj , cp^ . ... cp,. arbitrary functions re- 

 spectively of 2/i , u^ , ..., ?<,. , which also are definite functions oî x , y , z\ 

 thus <pi = (pi (»i) , (p3 = £p2 {vn) etc. 



If equ. (1) be differentiated partially Avith respect to x and y , we 

 obtain two equations containing the partial differential coefficients of z of the 

 first order together with the first derived functions of (pj , <p, , . . . «p^ • 

 Differentiating again, we get three equations with the three partial differen- 

 tial coefficients of z of the second order and the second derived functions 

 of (pi , . . . , (p,. , and so on. Proceeding thus , we get a system of equa- 

 tions, finishing with those, which contain the partial differential coefficients 

 of z of the n"' order and the n"' derived functions of (pi , (pj . . (p^ , « being 

 any whole number. These, joined with (1), are 1 -j- 2 -j- .. -\- (?i-|-l) = 



y^~r ^^ 'llELJ equations, containing the r arbitrary functions (p and their n 



first derived functions; in all {n-\-\)r functions (p , Çi'i (p'"^ That 



we may be able to get by elimination an equation fully freed from those 



functions, n must generally satisfy the condition *- "^ - *" ^ ^ (n-f-l)/--}-! , 



whence m > 2r — 1 . 



Nova Acta Eeg. Soo. Se. Ups. Ser. lU. 1 



