The Resultant of a Set of Homogeneous Lineo-Linear Equations. 375 
minant would be {m-{-n — l)\jn\{m--l)\ and the order of the deter- 
minants forming the elements would be n. In either case the degree of 
the resultant would thus be {m-\-n — l) \l(m — l) ! (n — l) !, as we know that 
it ought to be. 
3. Now, neither of these forms is the form which Sylvester considered 
satisfactory. In a paper " On a Theorem relating to Polar Umbrae,"" the 
nature of which may be more readily recognised by saying that it con- 
cerns a vanishing product of what Hankel afterwards called alternate 
numbers," he makes the remark that this is the fundamental theorem by 
aid of which I obtained the resultant of a lineo-linear system of equations 
in its most perfect form. It is easy to obtain two different solutions, each 
of them unsymmetrical in respect of the data of the question : the conver- 
sion and fusion of each of these into one and the same determinant, 
symmetrical in all its relations to the data, is effected instantaneously by 
a process derived from the above theorem." 
Doubtless the two solutions which Sylvester here depreciates are those 
which we have just given : but we are still kept in the dark as to the 
solution which he considered superior. 
4. Meanwhile Cayley had contributed to the Philosophical Magazine a 
theorem or two on so-called " canonic roots," which led Sylvester to 
publish a sequel: and in this sequel,! after referring to the contents of 
Cayley 's paper, he says: " This is the essence of the matter communicated 
by Mr. Cayley: but subsequent successive generalisations of the theorem 
have led me on, step by step, to the discovery of a vast general theory of 
double determinants, that is, resultants of bipartite lineo-linear equations, 
constituting, I venture to predict, the dawn of a new epoch in the history 
of modern algebra and the science of tactic." In the course of the 
account which follows we see how in his investigation on " canonic 
roots" a special set of lineo-linear equations (w, n = 3, 2) had turned up, 
why elimination was necessary, and how the idea of double determinants 
had arisen. Further, although the coefficients of the set of equations in 
question are not independent, being those of a persymmetrix matrix, we 
are able from an examination of the resultant obtained to guess what it 
would have been had the coefficients been perfectly general, like those we 
have used above, namely, 
a.b^c^d^l la.b^c^d^l-^^aAc^del \a,b^c^de\ 
a^b^c^d^ j I ajj^c^d^ ; - | a^b^c^de ) \ a^b^c^d^ | 
a.b^c^dc I I a^b^c^de I - 1 a^b^c^de \ \ a^b^c^d^, \ 
0. (C) 
* Proceed. R. Soc. London, xii. pp. 563-565; or Collected Math. Papers, ii. 
pp. 327-328. 
t Philos. Magazine, xxv. pp. 453-460 ; or Collected Math. Papers, ii. pp. 331-337. 
