The Adjugate of Bezout's Eliminant of Two Binary Quantics. 201 
It will be found, however, less stale and more effective to multiply the 
determinant 
-h 
1 
-K 
-h 
1 
-h 
«0 
columnwise by the 6-by-7 array 
234567 
2345678 i 
234567 
2345678 
a. 
for then we obtain 
«8 
\a.ybA 
the seven results of which give equivalents for all the /x's, and in four cases 
(/^4> /^5? /^6' l-^l) exact equivalents wanted. 
(4) What has just been established for the eliminant of the fourth order 
holds equally for any other order, and the mode of proof is quite general. 
In case of any doubt about the formation of the /it's, it need only be added 
that when the eliminant is of the n^^^ order, the row- numbers of are 
2, 3, 4, . . ., 2n — 1, and the column-numbers are obtained from 2, 3, 4, . . ., 
2t^ - 1, 2ri. 
(5) Le Paige's process, mentioned in § 3, led subsequent writers to the 
discovery of the fact that Bezout's eliminant is variously expressible as 
the result of multiplying n columns of Sylvester's eliminant by a trans- 
formation of the other 71 columns.* This suggests the making of a similar 
* So far as I am at present aware, the first publication of this is due to H. W. 
Tyler (' Sitzungst. d. phys -med Soc. zu Erlangen,' xxiii, pp. 33-128). He makes an 
oversight, however, in saying that the first set of n columns must be consecutive, 
and therefore in saying that the number of different pairs of factors is n -t- 1. 'J'he 
number in question is either n or 2n, according to the point of view. Thus, in the 
case where n is 3 there are the following equivalents of Bezouf s eliminant : 
654321 II 
123 II, 
I 465132 I 
I 234 :! , 
I 546^3 I 
I 845 il, 
the last two in the second column being evidently as worthy of enumeration as the 
first in the column. 
1 123456 j 
654321 !| 
1 123456 
! 123 
* 456 
1 456 
123456 
1 465132 II 
123456 
234 
• i 156 
1 156 
123456 
II 546213 ij 
! 123456 
345 1 
■ ' 126 
1 126 
