oil 
AS APPLICABLE TO TWO FUNCTIONS OF THE SAME DEGREE. 
which passes through the first coefficient of the first Bezoutic and the last coefficient 
of the last Bezoutic; and we may construct a quadratic homogeneous function of m new 
variables, such that its determinantive matrix shall coincide with the Bezoutic square 
so formed. This quadratic form may be considered in the light of a generating function. 
All its coefficients will be formed of quantities obtained by taking any two coefficients 
in one of the given functions, and two corresponding coefficients in the other given 
function, multiplying them in cross order, and taking the difference : each coefficient 
of the generating function in question will consist of one or more such differences, 
and will thus be of two dimensions altogether, being linear in respect to the coefficients 
of/, and also linear in respect to the coefficients of (p. This generating function I 
term the Bezoutiant, and it may be denoted by the symbol B(f, <p ) : the determinant 
of B is of course the resultant to / (p, and the matrix to B is the Bezoutic square to 
/, <p. Now we have seen that the decrease in the number of continuations of sign in 
the series 1, Bi(j:), B 2 (j?)...B,„/) (where Bi(a^), B2(jr)...B,„(/ are the (n) Bezoutics to 
/, (p), as X changes from a to h, measures the number of roots of fx retained in the 
effective scale of intercalations taken between the limits (a) and {b). If we take the 
entire scale between +oo and — oo the total number of effective intercalations will 
be the same, whether reckoned by the number of roots of / or of <p remaining; for 
these two numbers can never differ except by a unit, since no two of either can 
ever come together; but the number of each remaining in the effective scale will be 
ni—2i and m 2/ respectively, ^ being the number of pairs of imaginary roots and 
pairs of unseparated real roots of /and being the similar number for (p ; so that we 
must have i=zi'. 
Now obviously this number becomes measured by the number of continuations of 
sign in the signaletic series 1, (B,), (B2), ... (BJ, where in general (B,) denotes the 
principal coefficient in B,(jr). 
But (Bi), (B2), ... (B„) are the successive ascending coaxal minor determinants 
about the axis of symmetry to the Bezoutic square; and accordingly the number of 
continuations just spoken of, measures the number of positive terms in the Bezoutiant 
when linearly transformed, so as to contain only positive and negative squares, or in 
other words, measures the inertia of the Bezoutiant, the constant integer which 
adheies to it under all its real linear transformations. 
Art. (57.). This inertia is the same number as in the case of a homogeneous 
quadratic function of three variables, used to express a curve referred to trilinear 
coordinates, serves to determine whether such conic belongs to the impossible class 
or to the possible class of conics, being 3 or 0 in the former case, and 1 or 2 in the 
latter; or as in the case of a homogeneous quadratic function of four variables used 
to denote a surface referred to quadriplanar or tetrahedral coordinates, serves to 
determine whether such surface belongs to the impossible class or to the class con- 
sisting of the ellipsoid and the hyperboloid of two sheets (which are descriptively 
ndistinguishable), or to the hyperboloid of one sheet, being 0 or 4 in the first case, 
mdcccliii. q V ^ 
