DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 437 



Hence 



U 4 + 3U, 8 + 27u 4 e 4 = - 54w 8 (uQ' s + 2'Q 3 ), 



= -18u 8 (3uQ' 3 + 6'Q 3 ) 

 = -18 3 (3u8' s + 6t*'e 3 ), 



BO that U 4 is expressible in terras of invariants and covariants already retained. And 

 this relation between the invariants and covariants, viz., 



U 4 + 3U/ + 27^ + 18u 3 3 (M)! = 0. 



is practically the same as the differential equation, which may thus be considered as 

 replaced by a relation between its invariants and covariants. 



77. Taking now the case of the quantic of the fourth order, viz., 



(which we are entitled to include among the aggregate of invariants and covariants, 

 its index being ), we find, just as in the case of the equation, 



being ), we find, just as in the case of the equation, 



18w 3 u = 



so that U 4 can be expressed in terms of the invariants and covariants and of <J> V If, 

 then, 4> 4 be included as a fundamental covariant, and, in consequence of this inclusion, 

 all the proper derivatives from it be also included, then we have U 4 and all subse- 

 quent identical covariants expressible in terms of the covariants of the system thus. 

 increased. But if, on the other hand, the quantic (and derivatives from it) be not 

 included, then the number of the identical proper covariants may be taken as 

 unlimited ; and 4> 4 and all its derivatives are composite in terms of the invariants and 

 covariants. This is the reservation referred to in 67. 

 78. Taking, as a last example, the quantic, viz., 



(with 4> 6 = for the equation), we have 



4ttV - 30MU' V 

 U 8 = 4utt a 3u', 

 3 = 4w 2 u l " - 6t*uV + 3u', 



