DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 433 



the others can be expressed in terms of that one and of invariants. For from the 

 proper invariant B p there is derived the proper mixed covariant 



so that 



which verifies the statement. 



(ii) When the successive identical covariants U s , U 3 , U+, . . . are taken instead of 

 M, new covariants are obtained by forming the Jacobians of these covariants U and 

 the invariants Q. All such covariants are composite. For, taking U A with index /*, 

 equal to ^X (n 1), and denoting the Jacobiau of U A and O, by J, we have 



J = /JJ.e', - oe,U' A : 



and we have 



so that 



(n - 1) uJ + (re,U A + 1 = /xU A {(n - 



whence J is a composite covariant. Hence this class of covarianta must not be 

 included in the aggregate of proper covariants. (See also 39.) 



(iii) It is unnecessary to form the series of successive Jacobians from u and Q, (t<)j 

 as fundamental covariants ; for the Jacobian of u and 0, (u}^ is composite, and, there- 

 fore, all subsequent Jacobians are composite. To prove the statement, denoting this 

 Jacobian by 0, (w) 2 , we have 



= 2o-(2o- - n + 3) w /? 6,+ (n - 1) (4o- + 2) uu'%'. + (n - I) 2 u 



+ 2o-(n- 



after substitution. But 

 so that 



= ( - I) 2 (2o- -f 1 ) u 2 e' 2 , + 40 2 (2tr - n + 3) w^* 



+ (n - 1) 4<r (2<r + 1) uu'e^e', + 4^ (n - 1 ) 

 = 4ae^(n - 1) uu" - (n - 2) '*} 



+ (2<r + 1) { (n - I) 2 u 2 Q' 2 , + 4<r ( - 1 



and, therefore, 6, (n) z is composite. (See also 39.) 



MDCCCLXXXVIII. A. 3 K 



