26 M. Falk, On the Integration of partial 



gral ivill he (p =/(4) , / heing an arbitrary function. In this as well as 

 the following integrations the equations 



dzr-i,i = Zr-i+i.idx + ^r-i,i+x dy ..... (15) 



(i ^ , 1 , 2 , . , . r and r = , 1 , 2 , . . . , (ji — /)) may be ïised to 

 tra7isform and simplify. 



§■ 7. 



On partial differential equations of the form (12), where the 

 coefficients U and the right member V are functions of x and y only 

 or constants. 



We shall here prove some properties of these equations, which have 

 a remarkable affinity with those theorems, that hold for ordinary linear dif- 

 ferential equations of the n"' order. 



* Theorem 1. 



If u be any particidar solution of (12), then its general solution will 

 be obtained by adding to u the general solution of (12), provided its second 

 member be = 0. 



For putting ^: = tt + ^, whence ^„_/, , = ?<„_,,, -f C—.,, > the 

 equation (12) becomes 



g ^ «„-,•,. + Sc^C-.. = 





But j^ C/ «„_,,, = F, 



•=0 



because u is a solution of (12), therefore ^ must satisfy the differential- 

 equation 



1 = 



Q. E. D. 



