liKIUVATIVES ASSOCIATED WITH LINEAR DIFFKUKNTIAL EQUATIONS. 415 

 For (b), one of the simplest methods is to introduce the function 



and then, by 34, it follows that the quadriderivative of BJ^I is composite if that 

 of 4> Ai) . be composite, for the Jacobian of B A B (t and 4> A>(1 is expressible in terms of 

 invariants of the first three degrees. Since **, is of index unity, its quadri- 



derivative T is 



T = 



Now ( 39) the Jacobian of a priminvariant and Q^^i can be expressed in terms of 

 invariants of the first two degrees, and, therefore, the Jacobian P of a priminvariant 

 and to^t can be expressed in terms of invariants of the first two degrees; con- 

 sequently, the Jacobian Q of a priminvariant and of P can be expressed in terms of 

 invariants of the first three degrees at most. But 



Q = (rB,F - (<r + 2) PB; 



= o^X,. - 3oB,B>^ + **,, {"BX - (<r + 2)B' r 2 j , 

 so that 



Q + *** 

 Hence, 



2Q*,, + **, M B*. ! - 



and therefore T is composite. Hence also, the quadriderivative of B x>(1|1 is composite, 

 so that there is no proper quartinvariant among the functions (6). 



Combining our results, we see that the class (y) furnishes no proper quartinvariants. 



46. The general conclusion in regard to proper quartin variants is therefore the 

 following : 



There are n 2 independent and proper quartinvariants, and these are given by 

 (ix.); all other quartinvariants derived by these methods are either composed of 

 invariants of the three farmer classes, or, if proper, can be expressed in terms of in- 

 variants of the three former classes and of one or more of the n 2 independent proper 

 quartinvariants. 



Invariants of Higher Degrees. 



47. The investigation of the proper quiritinvariants proceeds on similar lines to 

 that for the quartinvariants. It is easy to see that, in forming the Jacobians of B Ci , 



* It may be remarked, as worthy of note, that T -f- 2* ! A,^ is the Schwarzian derivative with regard to 

 * of the absolute invariant log (6 A u e,,~ l> ). 



