2 M. Falk, On tue Integration op partial 



The least value , we can in ordinary cases give to n , is therefore 

 n = 2r—l. 



Thus 



A jn'imîtive (1), containing r arbitrari/ functions of the form, conside- 

 red above, tvill generally give rise to a partial differential equation of the 

 (2r — If order. 



But in special cases (1) may give a ditferential equation of an order 

 lower than the (2r — 1)''; for instance, when the equations, where the highest 

 ditferential coefficients of z are those of the »•"' order, do not contain the 

 functions (p , cp' , ... cp'^'"'-', but only (p/'', (p/\ ... (p',:' . Tliese equations are 

 (r-{-l) in number and will in general give by elimination of the functions 

 cp''''' a partial ditferential equation of the r"' order. If now r be > 2 , 7i 

 will be > r ; because n — r = r — 1 . If r = 1 , then n = 1 , and the 

 special will coincide with the general case. 



Without further reasoning we call (1) tlie general solution or primi- 

 tive to the ditferential equation, if (1) contains r arbitrary functions, the 

 order of the differential equation being r , that is : if n = r . Tliis will , in 

 fact, agree with the statement of Lagrange respecting partial differential 

 equations of the first and second order. It may be observed, that for r = 2 

 the special case jî = 2 (in the general one, n = 3) coincides with the state- 

 ment just cited. 



Some examples will throw light on the above theory. We will, how- 

 ever first make a few remarks as to some points in it. A solution of a 

 partial differential equation ought to be called general, Avhen it contains as 

 many distinct arbitrary functions, as possible, and satisfies the differential 

 equation. Now it may perhaps happen, that a given partial differential 

 equation of the r"' order cannot be satisfied by any relation containing 

 .T , y, z and r distinct arbitrary functions; the solution which in this case 

 would be called general, is then according to the definition above given, 

 not really so. Thus there may perhaps be partial differential equations, 

 deprived of any general solution in the usual sense of this term. — As to 

 the form of (1), it is not the most general possible, because each of the 

 functions (p contains only one definite function v , although it might have 

 contained any number of such functions; it is, still, enough for our purpose, 

 because, as will be seen afterwards, in this memoir we only obtain general 

 solutions of the form (1). 



