SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 495 
suggestion as to the mode in which the theory of Intercalations may hereafter be 
found to admit of being extended from a system of two general functions of .v, to 
a system of three general functions ot\v, i/, four general functions of .r, y, and in 
general to a system of s general functions of s— 1 variables, or which is the same thing, 
of s homogeneous functions of s variables. In the case of two functions of .r, f{x) 
and fx=0 and <px=0 may be considered to represent two systems of points in a 
1 ight line ; and the theory relates in this case to the relative positions of these two 
“ Kenothemes ” or point systems ; and of course using x and y to denote the distances 
ot any point in a line from two fixed points therein respectively, instead of fx and (px, 
we may employ two homogeneous functions of x and y, as f{x, y) and p (x, y), to 
denote these two systems of points. So, similarly, if we have three functions of two 
variables, /(cr, y), g{x, y), h{x, y), which I shall suppose to be of the same degree, we 
may consider the mutual relations of the Monothemes, that is to say, the three plane 
curves, denoted by the equations f{x, y) = 0, g{x, y) = 0 , h(x, y) = 0 . Now every two 
of these will intersect one another in a system of points, which we may call (f, g) for 
the intersections of/’and g, (g, h) for those of (g and A), and {h,f) for those of h and 
f. If we take any two of these systems of intersections, as (f. g) and (g, h), they will 
both lie upon one of the given curves (g). And by reading off the two systems of 
points (/, g) and (g, h), arranged according to the order upon which they are disposed 
upon the curve g, we may, by following the course of such curve, form a scale of 
effective intercalations for these two systems, and in like manner for the two systems 
(g, h) and (A,/ ) ; (A,/) and (/, g). Now I believe that it will be found that when 
y, g, A represent any algebraical curves consisting of a single continuous line, either 
extending to infinity in both directions, or returning to itself (and I have fully satisfied 
myself of the truth of this for the case of ellipses), each effective scale of intercalation 
will contain the same number of pairs of points ; if, however, the curves consist of 
moie than one branch, as if hyperbolm be considered, such is no longer necessarily 
the case ; from these facts, conjoined with the light thrown upon the subject by its 
relation to the theory of combinants explained in the succeeding section, I am induced 
to infer the probability of the truth of the following law (which, for avoidance of 
fuithei unceitainty, I confine to the case of functions of the same degree), viz. that 
if/, A be three homogeneous functions of x, y, and of the same degree, and if 
U, V, W be any three linear functions of/g. A, and if U=0, V=0, W=0 be treated 
as the equations to three cones, and if we form an effective scale of the intercalations 
of the lines of intersection of U and W, and V and W, according to the order in 
which they are disposed upon W (which seems to require that the lines shall be con- 
tinuous, in order to admit of a fixed order of reading off the intersections of any two 
of them upon the third) ; then whatever value may have been given to the coefficients 
in the linear functions the number of elements remaining in any such scale will (as I 
conjecture) be constant, and some theory (to be discovered) for three functions 
analogous to that of Bezoutian residues for two functions will serve to determine the 
MDCCCLIII. 3 T 
