418 MB. A. R. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT- 



the invariants @ 3 and 4 by a process which is effectively a continued repetition of 

 the Jacohian process ; and he has* two derived invariants, A, which is practically 

 BKIOSCHI'S quadriderivative of 6 3> and 8, practically the same function of @ 4 . He 

 also (1. c., p. 339) forms Jacobians, which can be expressed in terms of functions 

 B 3i r (in the notation of the present memoir) ; and these together constitute his 

 aggregate of invariants for the quartic. 



Lastly, the important simplification of the forms of the invariants due to the 

 reduction of the equation to its canonical form has been repeatedly remarked in the 

 preceding paragraphs ; it is, in fact, owing to this that the foregoing classification has 

 proved practicable. When, however, the differential equation is not assumed to be 

 thus reduced, a change necessarily takes place in the explicit forms of all the in- 

 variants ; thus, for instance, in the case of a non-evanescent coefficient Q 2 , it is not 

 difficult to verify that 



from which a non-canonical form of _ l the value of & ff being supposed known is 

 at once apparent. But into the expressions of those proper invariants which are 

 Jacobians the coefficient Q 2 does not explicitly enter until substitution begins to be 

 made for the invariants in this Jacobian form. 



Finality of the Results. 



51. The results so far obtained, though very general, have not been shown to be 

 exclusively so. It has been proved that all the linear invariants which exist are 

 included in the set of priminvariants ; and that all the invariants derived from them 

 by the given methods can be expressed in terms of the proper invariants of the classes 

 as arranged. But no proof has been given that, for degrees higher than the first, any 

 invariant possible can be deduced by the methods used, or that any invariant can be 

 expressed in terms of the assigned invariants. Until one of these two propositions 

 (or some equivalent proposition) is established, we are not in a position to declare that 

 all possible invariants of the differential equation can be expressed in terms of the 

 given invariants. 



The consideration of this question will be deferred until Section VIII., where the 

 investigation will include not merely the invariants, but other invariantive functions 

 yet to be obtained. 



* ' Acta Math.,' vol. 3, pp. 335 and 341 respectively. 



