28 M. Falk, On the Integration of partial 



Corollary. Applying theorem II to the equation (43), we see, that 



2 = Q Ç),. (?/ — m^x) also is a solution of it; and this solution is general, 



I— I 

 if the equ. (44) have no equal roots and all tlie arbitrary functions cp be 

 independent of each other. That this is the very same general solution, 

 that Avould be obtained by performing the integrations according to the fore- 

 going general theory, is easily shewn. For here (21) and (23) are imme- 

 diately integrable, as all the ytt's are constants and give n first integrals that 

 will be obtained from the equation: 



i=n-X 



1 = 1) 



by putting successively r = 1 , 2 , . . . , n . Hence we get the differential 

 coefficients of z of tlie (n — 1 )" order as sums of the functions 4 multi- 

 plied by certain constants. Substituting these expressions in (15) and inte- 

 grating, we get successively the differential coefficients of ^^ of the (n — 2)"'', 

 (n — 3)''' , . . . , 2'"' and first order all as linear functions of such arbitrary 

 functions; and, finally, substituting in dz = ^i ^ tZ« + c„ i dy and inte- 

 grating, : will also be expressed as a sum of n arbitrary distinct and in- 

 dependent functions. Q. E. D. 



Theorem IV. 



If a root m he repeated A times in (44), then z = 4 (j/ — mx) , 



z = œ-^i(y — ma), z = «^^-'O/ — mx) , , z = a'"-' 4*-i (v — m.c) 



are all solutions. 



For writing the equation (44) briefly /(»») = 0, we get, by sub- 

 stituting z = 4 (.'/ — "*^0 i" the left member of (43), the identity 



I ^'-fef^'' = (-!)■ 4" (.»-».-)/(».) ■ ■ ■ (45). 



Differentiating partially with respect to m, changing the sign of 

 every term and putting a new arbitrary function 4i instead of 4') ^^'e get 

 the new identity 



S ^'- ^dé^^— = (-l)"[-4'"^(3/-»^4/'(»0-4r' {y-mx)f{m)\..im). 



