376 
Transactions of the Boyal Society of South Africa. 
The superiority of this over the two others is at once apparent. It is not 
only that the ehminant is of lower order : there is the further advantage 
that the determinants which compose its elements are all primary minors 
of the given array of coefficients. 
If we use 
(a, /3, y, ... ) n, r, ...) 
to stand for 
am, [3m, ym, . . . , an, j3n, yn, ar, /3r, yr, . . . 
and denote a primary minor of an oblong array by the numbers of the 
columns which it occupies in the array, the result may be formulated 
thus : 
The resultant of the set of equations 
(a. 
... ae\{k, r), '0 (u,v) = 0 
... be^ 
... del 
1234 
1235 
1236 
1345-1246 1456 
2345- 1256 2456 
2346- 1356 3456 
= 0. 
A widely general result including this — in fact, the result for the case 
m, n — m, 2 — was given by me in 1905 before the conception of double 
determinants had been brought to my notice." 
5. So far as can be learned, Sylvester never returned to the subject. 
The question therefore remains as to how the result just given can be 
established. According to himself, he obtained it by a sort of condensa- 
tion-process performed on either of the other two : but what the exact 
nature of this process was it is hard to guess, and practically no help is 
got from being told that it was dependent on his theorem about polar 
umbrae. 
6. It is therefore all the more interesting to know that it can be 
obtained quite independently of the two others and as easily as either. 
For, writing the four given equations in the form 
ai^it, + a^vE + {a22i + a^v)rj + {a^u + a^v)^ — 0, 
and eliminating ut„ vl, t], 4, we have 
a2U + a^v 
CM + C/V 
* Tr ansae. R. Soc. 
pp. 118-121. 
Edinburgh, xlv 
'5^ 
pp. 
0, 
a^u + a^v 
h^u + h^v 
c^u + c^v 
d^u + d^v d^u 4- d(,v 
1-7 ; and Messenger oj Math., xxxv. 
