DIFFERENTIAL EQUATIONS OF THE n"' OrDER. 



11 



r = XY — XY\ 

 rj ^ Y'y — YY' 



... (13). 



Admitting tlie extreme values ^ i = , ^w.,, = 1 and ^„= , the ex- 

 pression for Î' gives also L\ and L\ . 





To integraie the differential equation (1'2) , when it has a first 

 integral of the form (3). 



The object of a first integration of (12) is to return from this dilie- 

 rential equation to a first integral of it of the form q =^f{'l) such as equ. 

 (3). In order to find a method of performing that integration we have to 

 shew, that the two simultaneous equations 



y = a, ip = ß (14), 



the conditions (11) being satisfied, will also satisfy the equation (12) and 

 reproduce it, if (14) be differentiated totally and the ditferentials elimina- 

 ted by means of the equations arising from the circumstance, that c is a 

 function of x and ?/ , viz.: 



dZr-ij=:r-Uyj dx -}- Zr-iJ+i^lJ (15), 



where Ave have to put {=0. 1, 2, ... r successively and then j- = 0, 1, 



2 , . . . , (n— 1) . 



Now (14) gives by difterentiation, if all ditferentials, except dx and 

 c?y, are eliminated by means of (1.5) 



i=n-l 



\ 



(16). 



dy 



and, if here we eliminate -^ , the equation (5) is reproduced, from which 



dx 



again by means of (11) the ditîerential equation (12) may be deduced. 



Upon this mode of deriving the differential equation from (14) a 

 method of obtaining the functions </■ and ip may now easily be founded ; 



