416 MR. A. R. FORSYTH ON" INVARIANTS, COVARIANTS, AND QUOT1ENT- 



and X for all values of cr and X, the same limitations on the mutual independence of 

 the (n 2) 2 functions so derived exist as in 40 and 42 ; and hence of this type 

 there are n 2 proper and independent quintinvariants given by 



ri ,e', ....... (x) 



the remaining proper quintinvariants being expressible in terms of the functions 6^4 

 and of invariants of the first four classes. 



48. By means of some of the results obtained we can show that all the invariants, 

 obtainable by any of the methods hitherto used or by any combination of them, are 

 expressible in terms of these different classes in succession of n 2 proper invariants 

 associated with the n 2 priminvariants. For 



(i) These proper invariants of any class are obtained by forming the fitting 

 Jacobians of the proper invariants of the class next preceding and the priminvariants ; 



(ii) By proposition (A) of 36 and the theorem of 44, it follows that all other 

 Jacobians are composite ; 



(iii) By the analysis of 45 it follows that the quadriderivative of any Jacobian 

 is composite, if we retain as representative invariants the successive Jacobians of 

 proper invariants. Now, after ,!, all the proper invariants ^.j, ,,,3, . . . are 

 Jacobians, and therefore quadriderivative functions formed from them are composite, 

 A result already proved in 45 for ^ , ; and thus the quadriderivative operation 

 applied to any proper invariant will produce only a composite invariant. 



(iv) It is easy to see that, if we take any proper invariant <I> of a class higher than 

 the first and from it, considered as a fundamental invariant, construct the same 

 functions as _ 2 , ,, 3, ... are of , all the resulting invariants will be composite. 

 For, considering in particular the cubiderivative function of <J> corresponding to a< z, it 

 will be the Jacobian of the invariant <> and of the quadriderivative of that invariant ; 

 this quadriderative will in general come under the he;id of those considered in (iii), 

 and therefore will be composite ; but in any case the theorem of 44 shows that the 

 function will be composite, since 3> is of a degree higher than the first. Similarly for 

 all the other functions. 



It therefore follows that the operations, similar to those whereby the invariants 

 <r, i. <r,2> are constructed from , only lead to composite invariants when 

 applied to proper invariants of any class beyond the first, and that the only operation 

 which can lead to proper invariants is the Jacobian, and even that operation onlv 

 produces proper invariants of any degree when applied to the n 2 invariants ,, and 

 the respective proper invariants of the preceding degree associated with r . 



49. The general conclusion as to the derived invariants is as follows : 



It is convenient to range the derived invariants in classes ; all the invariants in any 

 one class are, when the differential equation is taken in its canonical form, homogeneous 

 in the coefficients Q of the equation and their derivatives ; and the degree of any 



