408 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 and, therefore, we may take a new invariant 



;- (2o- +l)e^ 2 ....... (vi.) 



as the derived invariant in its canonical form. The index of 0,^ is evidently 2(<r + 1) ; 

 the number of these derived invariants is n 2 : and it will be convenient to call this 

 function of 6 ffi sometimes the quadriderivative function, sometimes the quadrinvariant, 

 associated with 0,. 



34. All these quadriderivative functions, derived from non-composite invariants, are 

 algebraically independent of one another ; but there is no a priori necessity that the 

 quadriderivative function of a composite invariant should be a composite function. 

 Let such an one be 



then the function obtained from it by the foregoing process is 



* pil = 2 (x + p.) <D P <*>; - (2\ + 2/t + 1) 



so that 



Hence, ^> PI I cannot be composed from the invariants ah-eady obtained ; but, if we 

 choose to introduce a new invariant 



the index of which is unity, then < PI , is composite. 



35. This new invariant can be otherwise derived. For / is an absolute invariant 

 (of index zero), and therefore 



