which occur in the Calculus of Invariants. 465 



In the further development of this theory, it will probably 

 be found expedient to suppose the number of the elements, 

 a, b, c, . . . j, k, I, to become finite, which will limit the number 

 of the derived extensors, and to study the mutual reactions of 

 the correlated series of extensors (with their derivatives), which 

 we may characterize respectively as the E and H series, where 





Either of the above two primitive forms (as it is the imperishable 

 glory of Professor Cayley to have discovered*) is sufficient in 

 itself for testing the nature of every invariant satisfying the 

 necessary and obvious condition of weight, and for deducing the 

 complete form of a covariant from either of its extreme terms ; 

 which latter consideration affords, I think, a sufficient ground for 

 the name (of some kind or another so much needed) Extensors, 

 which I propose to give to these too-long-suffered-to-remain 

 anonymous test operators and their derivatives. 



K, Woolwich Common, 

 November 13, 1866. 



Postscript. 



Since the above was sent to press it has occurred independently 

 to Professor Cayley, to whom I had communicated a sketch of 

 the theory, and to myself, that the general conclusions contained 

 in the text above would remain valid for a much more general 

 class of operants than those there defined; and there can be 

 little or no doubt that such is the case for all operants lineo- 

 linear in a set of elements a, b,c, . . ., and their prse-reciprocals 



-r3 -jji > -r , . . . . Moreover a material improvement in the no- 

 menclature has suggested itself, which I proceed to explain. It 

 is most important in this theory to be able to distinguish 

 between the corpus or root of an operator viewed as a function 

 and the operator itself, and to be in possession of a single 



* But this magnificent discovery, whereby the determination of the 

 number of fundamental invariants to a binary quantic of a given degree 

 is reduced to a problem in the partition of numbers, it is but justice to 

 M. Hermite to state, took its rise in that great analyst's discovery of the 

 octodecimal invariant of the binary quintic. So long as the existence of 

 this fourth invariant to that form was unsuspected, it must have remained 

 impossible to conjecture the sufficiency of the single partial differential 

 equation-test. 



Phil. Mag. S. 4. Vol. 32. No. 218. Dec. 1866. 2 H 



