UERIVATIVKS ASSOCIATED WITU LINEAR DIFFERENTIAL EQUATIONS. 429 



so that we may consider u as a covariant of index i ( 1). Proceeding in the 

 same manner as in 33, it is easily found that 



which at once gives a new invariantive form. When the transformed equation is in 

 its canonical form, so that Q 2 vanishes, the new covariant is 



with index 2 ; or we may write 



U 8 = (n - 1) uti" - (n - 2) t*' . . . . . . . . (xvi.) 



of index 3 n. This is the quadriderivative of u.. 



65. There are thus two co variants u and U 2 ; from them as fundamental co variants 

 we can deduce the series of successive Jacobiana. Thus the covariant next in degree 

 is (40) 



or say it is 



U 8 = (n - 1) uU' 2 - 2 (n - 3) U a u' ........ (xvii.) 



with index 3 n ^ (n 1) + I = f (n 3). The next is 



U 4 = (n l)tiU' s 3(n 3) U,u' ....... (xviii.) 



with index f (74 3) (n 1) + 1 = 2 (n 3) ; and so on. And the rth 

 covariaut in the complete succession is 



U r = (ft - 1 ) u U r '_! - (r -l)(n- 3) U r _! u' . . . (xix.) 



with index ^ r(n 1). By means of the propositions used for the invariants it is 

 easy to see that this series constitutes the aggregate of proper covariants involving u 

 alone ; for all others, obtained by combinations and by the application of the functional 

 operations to combinations other than those which give results (xvi.)-(xix.), are, by 

 those propositions, proved to be composite. 



