414 MR. SYLVESTER ON A THEORY OF THE CONJUGATE 
shown, this is not a property peculiar to the Bezoutiant ; in fact there exists a whole 
family of quadratic forms of m variables, lineo-linear (like the Bezoutiant) in respect 
of the coefficients in /"and <p, all of which enjoy the same property. Ihe number of 
individuals of such family must evidently be infinite, because any linear combination 
of any two of them must possess a similar property ; I have discovered, however, that 
the number of independent forms of this kind is limited, being equal to the number 
of odd integers not greater than the degree of the two functions and (p. In arts. 
(67.) and (68.), I give the means of constructing the scale of forms, which I term the 
constituent or fundamental scale, of which all others of the kind are merely numerico- 
linear combinations. This scale does not directly include the Bezoutiant within it, 
and it becomes an object of interest to determine the numbers which connect the 
Bezoutiant with the fundamental forms; this calculation I have carried on (in aits. 
(69.), (TO*); (71*)) f^’om m~l to m~G inclusive, and added an easy method of con- 
tinuing indefinitely. In this method the numbers in the linear equation corresponding 
to any value of m are determined successively, and each made subject to a veiification 
before the next is determined, there being always pairs of equation-s which ought to 
bring out the same result for each coefficient. 
In the next and concluding art. (72.), I remark upon the different directions in 
which a generalization may be sought of the subject-matter of the ideas involved in 
M. Sturm’s theorem, and of which the most promising is, in my opinion, that which 
leads through the theory of intercalations. Some of the theoiems given bj me in 
this paper have been enunciated by me many years ago, but the demonstrations have 
not been published, nor have they ever before been put together and embodied in that 
compact and organic order in which they are arranged in this memoir, the fiuit of 
much thought and patient toil, which I have now the honour of presenting to the 
Royal Society. 
June 16, 1853. 
In a supplemental part to the third section I have given expressions in terms of the 
roots of px and fx for the quotients which arise in developing ^ under the form of a 
continued fraction, and some remarkable properties concerning these quotients. In a 
supplemental part to the fourth section I have given an extended theorp of mp new 
method of finding limits to the real roots of anp algebraical equation. This method, so 
extended, possesses a marked feature of distinction from all preceding methods used for 
the same purpose, inasmuch as it admits in everp case of the limits being brought up 
into actual coincidence with the extreme roots, whereas in other methods a wide and 
arbitrarp interval is in general necessarilp left between the roots and the limits. 
