^ 
428 MR. SYLVESTER ON THE RESIDUES OF EXPANDED 
m3, &c. being functions of the coefficients only of U and V ; and it is not 
without interest to observe (which is capable of an easy demonstration) that the 
simplified residues contained in R„ R^, &c., found according to this mode of develop- 
ment, will be the successive dialytic resultants obtained by eliminating the (i-l)th 
highest powers of x between the < first of the system of the annexed equations (sup- 
posed to be expressed in terms of x) 
U=0 
v=o 
0? . iJ = 0 
j;. v=o 
u = o 
V = 0 
&c. &c. 
If we combine together of the above equations, the highest power of x enteiing 
on the left-hand side will be and we shall be able to eliminate 2/ of these factors, 
leaving the highest power remaining uneliminated. If we take 2 i, i.e. i pairs ot 
the equations, the highest power of x appearing m any of them will be h and we 
shall be able to eliminate between them so as still to leave ^ ^\i.e.x” ‘asbetore, 
the highest power of x remaining iinelnninated ; and it will be readily seen that such 
of the simplified residues corresponding to this mode of development as occupy the 
odd places in the series of such residues, will be identical with the successive simplified 
residues resulting from the ordinary mode of developing y under the form ot a con- 
tinued fraction. 
Art. (11.). It has been shown that the simplified residues ot fx and resulting 
from the process of continued division are identical in point of form with the 
secondary Bezoutians of these functions, but it remains to assign the numerical 
relations between any such residue and the corresponding secondary. 
To determine this numerical relation, it will of course be sufficient to compare the 
magnitude of the coefficient of any one power of x in the one, with that ot the same 
power in the other ; and for this purpose 1 shall make choice of the leading coefficients 
in each. In what follows, and throughout this paper, it will always be understood that 
in calculating the determinant corresponding to any square the product ot the terms 
situated in the diagonal descending from left to right will always be taken with 
the positive sign, which convention will serve to determine the sign of all the other 
products entering into such determinant. Now adopting the umbral notation for 
determinants*, we have, by virtue of a much more general theorem for compound 
See London and Edinburgh Philosophical Magazine, April 1851. 
