50 Prof. Sylvester on the Multiplication of 



limited to the case where those two operants, <f>, ty, are each of 

 them linear quantics in regard of 8 X} 8 P , B gi . . . The proposition 

 advanced guardedly in the Postscript concerning any lineo-linear 

 functions of a?, y, z, . . . 8 X , 8y, 8 Z , . . . (" there can be little or no 

 doubt, &c") I now also wish to be understood as affirming ab- 

 solutely. I proceed to give a universal theorem for the mul- 

 tiplication of any number of operators, energized functions 

 of x,y,z, ... ; 8 X , 8 y , 8 S , . . . freed from all restriction as to 

 linearity of form in respect to the latter set. 



The method by which I arrived at this very general theorem 

 was in substance identical with that embodied in the demonstra- 

 tion spontaneously furnished me by my ever ready correspondent 

 Professor Cayley ; and as I cannot improve upon his statement, 

 it would be a waste of time to substitute my own words for his. 

 Accordingly, after enunciating the theorem, I shall give the proof 

 of it in the very words of our unrivalled Cambridge Professor, 

 from which it will be seen that in essence this theorem consists 



soon as it becomes necessary (as will probably before long be the case) to 

 express the specific relation of the star to something which limits and dis- 

 criminates its mode of application, it must in its turn develope into a third 

 species of symbolical quantity; and so there may be in store for the future 

 of algebra an endless procession of more and more abstract symbols of 

 operation, each successively developing into a more and more subtle species 

 of quantity, suggesting the analogy of successive stages of so-called impon- 

 derability in the material world. 



A propos of Arbogastiants, it is worthy of a passing notice that if I be 

 any invariant to the form (a, b, c, . . .h, k), and we write A for the Arbo- 



(A*) n 

 gastiant (I8j c j r2kdh + • • . ), then ~fTZr expresses the effect of the substi- 



fb, c, . . . k, I "I 



h 2- P er f° rme d upon I. This theorem is an easy conse- 



quence of the conjunction of the three circumstances, (1) that if \ x , y is 

 what I becomes when for a, b, c, . ..k we substitute respectively ax-\-by, 

 bx-\-cy, ... kx-^-ly, I x , y will be a covariant to the form (a, b, c, . . . h, k, I), 



(A*)« 

 and that consequently the last coefficient in I lt , y will be — — I ; (2) that 



this coefficient must bear the same relation to L k, . . .c,b as the first does 

 to a, b, c, . . . k ; and (3) that an invariant to the form {I, k, . . <c,b) is iden- 

 tical with the same invariant to the form (b, c, . . .k, I). 



I think I have been informed that Leibnitz was the first to employ the 

 method of the so-called separation of symbols : in his tract on the Calculus 

 of Differences, the poet sage of Collingwood contributed powerfully to 

 its further development ; if he should chance to cast his eyes over these 

 pages he will, I fear, stand aghast at the Frankenstein he has thus (it may 

 be unwittingly) played no unimportant part in bringing into existence ; or, 

 rather, I should fear, did not all the world know his perfect candour and un- 

 stinted sympathy with every form of manifestation of human intelligence. 



