DIFFERENTIAL EQUATIONS OF THE n"' ORDER. .7 



equations involving z and its partial ditferential coefficients up to tlie n"' or- 

 der, n being- the order of the ditferential equation, arising from the elimi- 

 nation of the arbitrary constants, we must determine it from the conditions 



(n+l)( 7t+2) > ,, , 1 ^ > njn^l) 



2 - -r , = . 



Thus when 1 < r < 2 , 



3 < r £ 5 , 

 6 < r < 9 , 



„ 10 < r < 14 , 



For the sake of the utmost generality it seems best to call (h) a complete 

 primitive, only when r = ^^ — li— — / — 1 , n being the order of the 



differential equation; this also agrees with Lagrange's view, for he supposes 

 r = 2 when n = 1 . 



Finally, as our resources for the integration of given partial diffe- 

 rential equations by obtaining their general solutions are restricted, we so- 

 metimes get solutions involving both arbitrary functions and arbitrary con- 

 stants; such solutions we will term mixed. Supposing 



z = f{x , r/ , Ol , a^ , . . . a^ , q^ , ,f^ , ... <f^) 

 a mixed primitive, then in general tJie order n of the resulting differential 

 equation will be determined from the conditions 



(/i-f-l)(/i-|-2) -. ,/,1X11 , -^ n(n\-l) 



^ o -^-^ > p + (Ji+l) '7+ 1 , p -\-nq > -^—^ . 



We consider in the same way the mixed solution as particular, when 

 the condition 



(!i±l|'it^ > p + (n + l) , + 1 



is satisfied. 



