DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 473 



respectively, where all the constants are arbitrary. Hence, from the point of view of 

 an integral equation, we may write 



where E, B, = A,, and so E, is an arbitrary constant. It therefore follows that 



[(*.-E.,2)].= 

 is a general first integral of 



[ <Ui = 0. 

 Again 



from the point of view of an integral equation, the value of F rt being D^/Cg, so that it 

 is an arbitrary constant ; hence 



=o 



V z ._ t 



is a general first integral of 



[(o-- *)]. = 0. 



Combining these results, we see that 



is a general second integral of 



[*., *]. = 0, 

 where 



1 



r. + fltv 



Similarly 



2, ] a = 



is a general fourth integral of the same equation, where 



1 



. = E. + 



F _L 2J3 , 



and so on. 



Similar results are obtainable for the equation which involves the derivative of even 

 order. 



MDCCCLXXXVI1I. A. 3 P 



