Fundamental Theorem of Invariants. 185 



intersect in a common point the centre of gravity of the whole 

 body of aesthetic ; but what that centre is or must be I have 

 not had time to think out. 



Johns Hopkins University, Baltimore, 

 November 13, 1877. 



Postscript. — In the first fervour of a new conception, I fear 

 that in the manuscript which is now on its way to England I 

 may have expressed myself with some want of clearness or 

 precision on the subject of pure and mixed differentials. I 

 will therefore add a few more explanatory and vaticinatory 

 words on this subject, through the medium of which I catch a 

 glimpse of the possibility of obtaining a simple proof of Gordan's 

 theorem, just as through the medium of pure differentiants 

 taken per se I caught a glimpse (almost immediately afterwards 

 to be converted into a certainty) of the proof of Cayley's theo- 

 rem given in this memoir. I conceive that what the ensemble 

 of pure differentiants have done for the one, the larger en- 

 semble of all sorts of differentiants, pure and mixed, taken 

 together, will enable me or some one else to accomplish for 

 the other. 



Any function of the coefficients of a quantic which is nulli- 

 fied by the operation upon it of H, which we may call the 

 revector symbol, or in other words, whose first revect is zero, 

 is a pure differential in x. Bo, of course, if nullified by the 

 operation upon it of 0, which may be called the provector 

 symbol, it is a pure differential in y. We may call ij— 2w } 

 where i is the degree of the quantic, j the order of a pure dif- 

 ferential, and w its weight in x y the grade of the differential, 

 and denote this grade by 8. 



The 8th provect of a pure differentiant in x is of course a 

 pure differentiant in y, which is 8 removes from x, as the pure 

 differentiant in x is 8 removes from y. If q be less than 8, the 

 ^th provect of a pure differentiant in x is a mixed differentiant 

 q removes from x, or, if we like to say so, (8—q) removes 

 from y. The grade of a mixed differentiant may be defined to 

 be the same as that of the pure differentiant of which it is a 

 provect or revect. 



Then, in the first place, we have this proposition : — If any 

 linear substitution whatever be impressed in the variables of 

 a quantic, the transformed value of any of its differentiants 

 will separate into two factors, of which one will be the deter- 

 minant of substitution raised to the power w, where w is the 

 weight corresponding to the order and grade of the differen- 

 tiant and the degree of the quantic. The remaining factor 

 will be a function of the coefficients of substitution, and may 



