DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 411 

 and therefore 



I*-- 1 ,, H "' 



' 1 ^ 



H A ^ 



Hence it follows the Jacobian under consideration can be constructed from prim- 

 invariants and proper quadrinvariants ; it is, therefore, a composite function, and 

 must be omitted from the aggregate of proper cubinvariants. 



40. There thus remain only the Jacobians of the priminvariants with the quadri- 

 derivative quadrinvariants, and of these the total number is (n 2) 8 . But, denoting 

 the Jacobian of 8, ( , and 8 X by , ? A , we have 



*,, = X8 A 8;,, - 2(0- + 1)8,,, 

 %. M = ft 6,e;, , - 2 (a- + i) 8,, , 



so that 



A - *e**r = 2 r + i e, 



and, therefore, when any invariant , iX is considered as given, any other of this type, 

 derived through 8, f t and so involving the same <r, can be expressed in terms of r , x 

 and of invariants of earlier classes ; and hence out of the n 2 functions derived 

 through 6,, , it is necessary to retain only one of them. This being so, it appears 

 natural to retain that function, which has X = or, and is the Jacobian of 6,. i and of 

 the priminvariunt 6,, with which 6,,, , is associated. Denoting it by 8,, 2 , we have 



;,i-(2o-+2)e,, 1 e;, ...... (viii.) 



with index 3cr + 3 ; the number of these functions is n 2, and their aggregate con- 

 stitutes the aggregate of independent proper cubinvariants. But it should be 

 remembered that there are (n 2) (n 3) other proper cubinvariants, which for their 

 expression require some one at least of this aggregate. 



The proper cubin variant 8, i2 will be said to be associated with 8^ 



3 a 2 



