52 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where S is Sylvester s bigradient. This consists simply in taking the 
1 +n-\-m equations 
and deducing 
= 
^m+n—1 d" C m+n _ 2 X + C 
/y»2 
m+n—Z' v 
+ • • 
. . \ 
A | 
G, m + 
+ 
V 2« 2 
+ • • 
xA = 
a m x 
+ 
+ . • 
x 2 A = 
a m x 2 
+ . . 
B = 
b n + b n _ x X 
+ 
b n _ 2 x 2 
+ . . 
II . 
b n x 
b r 2 
u n- !•*' 
+ . . 
A 
xA 
^m+n — 1 ^ m+n — 2 ^ m+n — 
ci m a m _i d m _2 
• eL m et m _ i 
0. 
B 
xB 
b n - 1 
b n 
b n - 2 
+n+m 
Jacobi’s theorem of 1835 regarding Bezout’s condensed eliminant 
suggests the similar theorem regarding the bigradient eliminant,* namely, 
if w be a common root of the equations 
a 0 x m + a Y x m ~ x + • • • = 0, b Q x n + bp? 1 - 1 + • • • = 0 , 
then the signed primary minors associated with any row of 
(a 0 , , a m ) n 
are proportional to 
w m+n ~\ W m+n ~\ ... ,10, 1 . 
In dealing with the highest-common-factor of A and B and with the 
subject of elimination Baltzer profits far less than he ought to have done 
from the work of Trudi, whom indeed he does not mention. 
Isic, E. (1873): Janni, V. (1874). 
[Sul grado della risultante. Giornale di Mat., xi. p. 253.] 
[Sul grado dell’ eliminante del sistema di due equazioni. Giornale di 
Mat., xii. p. 27.] 
* Gordan (1870) in quoting the two from Baltzer says that mn of the primary minors 
of the former eliminant are secondary minors of the latter. , {Math. Annalen , iii. p. 356.) 
