RELATIONS OF TWO ALGEBRAICAL FUNCTIONS. 
413 
changes will be lost or gained as x passes from positive infinity to negative infinity. In 
art. (50.) I observe that the theory of real roots of a single function given by M. Sturm’s 
theorem is a corollary to this theory of the intercalations of real roots of two functions, 
depending upon the well-known law, that odd groups of the limiting function /.r lie 
between every two consecutive real roots of fx. In art. (51.) I verify the law of reci- 
procity, already stated to exist between the residues of J^x and <px, by an h posteriori 
method founded on the theory of intercalations. In arts. (52.), (53.), (54.), I obtain a 
remarkable rule, founded upon the process of common measure, for finding a superior 
and inferior limit in an infinite variety of ways to the roots of any given function. 
This method stands in a singular relation of contrast to those previously known. All 
previous methods (including those derived through Newton’s Rule) proceed upon 
the idea of treating the function whose roots are to be limited as made up of the 
sum of parts, each of which retains a constant sign for all values of the variable 
external to the quantities which are to be shown to limit the roots. My method, on 
the other hand, proceeds upon the idea of treating the function as the product of 
factors retaining a constant sign for such values of the variable. In art. (55.), the 
concluding article of the fourth section, I point out a conceivable mode in which the 
theory of intercalations may be extended to systems of three or more functions. 
In Section V. arts. (56.), (57.), I show how the total number of effective inter- 
calations between the roots of two functions of the same degree is given by the 
inertia of that quadratic form which we agreed to term the Bezoutiant to /and 
and in the following article (58.) the result is extended to embrace the case contem- 
plated in M. Sturm’s theorem ; that is to say, I show, that on replacing the function 
of X by a homogeneous function of x and y, the Bezoutiant to the two functions, 
which are respectively the differential derivatives of f with respect to x and with 
respect to y, will serve to determine by its form or inertia the total number of real 
roots and of equal roots in/(.r). The subject is pursued in the following arts. (59.), 
(60.). The concluding portion of this section is devoted to a consideration of the 
properties of the Bezoutiant under a purely morphological point of view; for this 
purpose/ and cp are treated as homogeneous functions of two variables x, y, instead of 
being regarded as functions of x alone. In arts. (61.), (62.), (63.), it is proved that the 
Bezoutiant is an invariantive function of the functions from which it is derived ; and 
in art. (64.) the important remark is added, that it is an invariant of that particular 
class to which I have given the name of Combinants, which have the property of 
remaining unaltered, not only for linear transformations of the variables, but also for 
linear combinations of the functions containing the variables, possessing thus a 
character of double invariability. In arts. (65.), (66.), I consider the relation of the 
Bezoutiant to the differential determinant, so called by Jacobi, but which for greater 
brevity I call the Jacobian. On proper substitutions being made in the Bezoutiant 
of the {m) variables which it contains (m being the degree in x, y oi f and <p), the 
Bezoutiant becomes identical with the Jacobian to / and p ; but as it is afterwards 
