1876.] Clair aut Ian Functions and Equations. 45 



also that those of zero class (7l = 0) are 



.... "U„,.=^^V"^- (^) 



It is easy also to establish by expansion and comparison of coefhcients 

 that 



^ r, n — -^/j^rO ^P.^-r+p' ^ n, n — P, P> V / 



4. Differential ])roperties. — It is easily established by actual differenti- 

 ation that 



D^/U,,„=^-^U,,„+^ (7a) 



Hence also, by the theory of " general differentiation," 



D'-".*U,,„=''IT,,^,._,, 



whatever be the value of h (omitting all arbitrary terms). 

 These results may be thus expressed in words : — 



1°. Simple differentiation dejpresses the class, and raises the order^\^ 

 of a Clairautian, without affecting its rank. 



2°. All Clairautians of same class (Jc) and order (n), after h 

 differentiations (in the general sense), contain, omitting 

 arbitrary terms, a common factor y^^^''\ and are there- )-(8) 

 fore A.-th integrals of y^^+"'>. 



3°. Hence also all the above quantities may be expressed as 



integrals of the last of them, °U^^ j,^,,, or of — . 3/^^+" 



5. Symbolic forms. — It is obvious from results (7), (8), that (omitting 

 arbitrary terms) 



;^'^D\^H,^,,=^^=|^\^^^^^^ ... (9) 



Hence, by the known theorem — 



