412 Proceedings of the Royal Society of Edinburgh. [Sess. 
Sylvester, J. J. (1851, May). 
[Essay on Canonical Forms : Supplement to a “ Sketch of a Memoir 
on Elimination, Transformation, and Canonical Forms,” 36 pp., 
London. Or Collected Math. Papers, i. pp. 203-216.] 
In giving a preliminary notice of his general method for reducing odd- 
degreed functions to their canonical form, Sylvester says he based his 
method on the proposition that every one of the ^-line minor determinants 
of the array 
Ti T 2 T 3 . . . T n+1 
T 2 T 3 T 4 . . . T n+2 
T T T T 
x 3 x 4 -*-5 • • • A n+3 
T T T T 
-*-n -*-71+1 - L n+2 • • • -*-2 n 
a r-i b s+l + a r-i b s+i + p . . +a r n Z\b S +! 1 . 
vanishes if 
T i & 
This, which he hastily calls “ a beautiful and striking theorem,” and which 
he generalises in Note B of an Appendix, arises from the simple fact that 
each determinant is the product of two zeros, T* being 
(< \ <4 
;_i, o % +i , • • • , k ±\, o). 
It is of more importance, therefore, to recall that it was in this year 
that Sylvester made the fruitful observation, already chronicled,* that the 
determinants ac — b 2 , ace-\-2bcd — ae 2 — bd 2 — c 2 , .... are expressible as 
“ commutants,” or rather that these special determinants could be repre- 
sented in the umbral notation by using umbrae not wholly unconnected 
with one another. Thus, while 
00 
01 
02 
stands for the general determinant 
10 
11 
12 
20 
21 
22 
so long as the umbrae are understood to be entirely independent, it might 
also be used to stand for the special determinant 
00 01 02 
01 02 03 
02 03 04 
if some mark were added to indicate that in the development 01 is to be 
put for 10, 02 for 20 or 11, 03 for 12 or 21, and 04 for 22. 
* Proc. Roy. Soc. Edin., xxv. pp. 939-942. 
