DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 487 



(i) The r identical covarianta in u, given by u, U z , U 3 , . . ., U r (or their 



functional equivalents, reduced as in 132, 133). 

 (ii) The r f identical covariants in v p , given by v p , V, jZ , V^,, . . ., V^ (or their 



similarly reduced functional equivalents). This is the case for each of the 



associate variables v z , v s , . . ., v,-i 

 (iii) The derived invariants involving 0. alone and given by , 8^j, O^j, 



<#V,4, , &.* This is the case for each of the priminvariants 6 8 , B v 



> 8' 

 (iv) The mutually independent bilinear Jacobians ; as a set of algebraically 



independent functions, retained after the indications of 36, 72, we may 



take the Jacobian of 6 8 with each of the quantities u, v 3 v,_ l , S v 



. . ., e,. The total number of these is 1 -f (n 2) + (n 3), t.e., it is 



2n 4. 



Hence the total number of algebraically independent concomitants, involving the 

 specified quantities and obtained by our earlier methods, is 



pmn 1 P ' 



= r + 2 r f + 2 * M + 2 4 



jj=S M = J 



= N; 



and from each of them an integral can be constructed, which is independent of all the 

 other integrals. 



From the first three of the classes we have already had examples of the method of 

 construction of integrals ; as an example of the last class, we may take the subsidiary 



equation 



du 



(n l)tt 2/X6,. 



Previous integrals are u = A, 8^ = B, so that an integral of the equation which 

 appears is 



2/iBw' + (n 1) A8^ = C, 

 that is, 



2/x8^' -f (n 1) uS'^ = C, 



or, what is the same thing, 



It may also be remarked that, while the class (i) of functions constitutes the set of 

 integrals derived from the u fractions alone in the subsidiary equations, the class (ii) 

 constitutes the separate sets from the fractions in each of the other associate 

 variables taken individually and alone, and the class (iii) constitutes the set from the 



