534 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
the correspondent system of Co-bezoutiants for all values of m between 1 and 6 
under a synoptical form. 
m— 1 B=:Gi 
/y^ = 2 B=:2Gi 
m=3 B=3Gi— 4 G 3 
m = 4 B = 4 Gi+-^G3 
50 2 
m—h B = 5Gi+^G3+^G5 
m=6 B^GGi+YGa+y Grg. 
These series could if wanted be easily extended, and the calculation of the coefficients 
reduced to a mere mechanical procedure. 
If we suppose 7n to be or 2i — -1, we have the equation 
B=Ci.Gi+C 3.G3+ ...-f-Cji-i G2i_i ; 
and it appears from the foregoing instances that the comparison of the coefficients, 
either of u\, or of u^.u^ on the two sides of the equation, will serve to give Cj and 
(which is always m being known), C3 may be found by a comparison of the coeffi- 
cients either of u^.u^, or of and so on for C5...c-2i_i ; all the coefficients in the 
equation for B above given, thus admitting of being found separately and successively 
and in two modes, so that there is a check at each step upon the correctness of the 
computations : the only exception to this last remark is (when m is odd) foi the last 
coefficient of which the above condensed method affords only a single determination. 
I need hardly add the remark, that in substituting a:™ ; ...x.y ...y in 
place of u,, u,, respectively, all the G’s become (to a numerical factor 
pres) identical with one another and with the Jacobian to the system (/?>)■ 
Art. (72.). The foregoing theory took its origin (as will have been readily imagined) 
in meditations growing out of the celebrated theorem ot M. Sturm. There appear 
to be several directions in which a development or extension of the subject raattei of 
that theorem may be sought for. Thus a theory may be constructed relative to a 
single function of one or more variables, viewed in all cases as representing a geome- 
trical locus. In the limiting case, when this locus becomes a system of points in a 
right line, we have the theorem of Sturm ; generally the theory will be that of con- 
tours. Or, again, a theory may be formed in which the number of functions is 
always kept equal to that of the variables. We have then a theory of discreet points 
corresponding to roots, the number of real ones of which comprised within given limits 
it is the object of such theory to determine. M. Hermite, in a memoir recently pie- 
sented to the French Institute, appears to have made a valuable addition to the^ 
Sturmian theory extended in this direction, to which the beautiful researches 0 
M. Cauchy and the joint labours of MM. Liouville and Sturm, with reference to 
