(199) 
NOTE ON THE ADJUOATE OF BEZOUT'S ELIMINANT 
OF TWO BINAEY QUANTICS. 
By Sir Thomas Muir, LL.D. 
(1) It is manifest that the two equations 
hQX'^ + h^x^ + b.,x- 4- h.^x + h^ — 0^ 
give immediate rise to the four 
h^^x^ -\- h^x + he, h.^x H- ?>4 
ao^e^ + a^x^-\- a.-,x + a., a. 
= O, 
= 0, 
= o, 
= 0: 
0, 
and that these when expressed explicitly as cubics in x furnish us at once 
with 
the left-hand member of which is known as Bezout's eliminant. 
(2) To the adjugate of this eliminant considerable study has been given 
since Jacobi first drew attention to it ('Crelle's Journ.," xv, pp. 101-124). 
Unfortunately there has been no simple mode of expressing its elements, 
which rapidly increase in complexity with the degree of the determinant. 
Taking advantage of the fact much later established that the primary 
minors of Bezout's eliminant have equivalents among the secondary minors 
of Sylvester's eliminant, I have succeeded in obtaining for the adjugate an 
expression whose law of formation is perfectly simple. Before stating it, 
it is necessary to recall the fact that the most convenient form of Sylvester's 
eliminant is that in which the rows of 5's follow the opposite order of the 
rows of a's, being in the case which we are considering 
