496 MB. SYLVESTER ON A DEVELOPMENT OF THE METHOD OF ASSIGNING 
number of the elements so remaining. And so, in like manner, but rvith a difficnlty 
increasing at each step (as at the next step we should have to pass mto pias, space ot 
four dimensions), a theory of intercalations may be conjectured to exist for any (u) 
general functions of any (w— 1) variables. 
Development of the method of assigning a superior and inferior limit to the roots of any 
algebraical equation. 
Art. (».). Since the articles in the preceding part of this section on the method of 
discovering limits to the roots of an algebraical equation were written, the method ot 
which the germ is therein contained has presented itself in a much more fully deve- 
loped form, which I proceed to exhibit : for greater simplicity I shall suppose to 
be of H-1, and fx to be of n dimensions in x, and that by means of the ordinary- 
process for common measure (except that as in Sturm’s theorem the sign of all the 
remainders are changed) | has been thrown under the form of the improper con- 
tinned fraction 
j 1 l_ 
< 1 \— 
where a,m...?.are all restricted to signify simple linear functions of x. 
Suppose the series q„ q„ ...?» to be resolved into the distinct sequences 
qi+i 
in such a manner that in each sequence as qi+, qi+.-.-q, the coefficients of x have all 
the same sign, but that in any two adjoining sequences the coefficients of a’ have 
opposite signs, so that for instance in qi and ^,+1 the coefficients of o’ are unlike, as 
also in q^ and q^a ,^ ; there will of course be nothing to preclude any of these sequences 
becoming reduced to a single term. 
The first theorem is, that the greatest and least roots of the product ot the ciinni- 
lants ^ ^ -I r „ -i 
are superior and inferior limits to the roots oi fx. To prove this theorem I begin 
with premising the two following lemmas, one virtually and the other expressly 
contained in the Philosophical Magazine for the months of Septemher and October 
of the present year*. 
* Each of these two lemmata flows readily from the faculty previously adverted to engaged by evei} cumulant 
of being representable under the form of a determinant. As to the second lemma, it becomes apparent imme- 
diately when the cumulant is so represented, by separating the matrix into two rectangles an expressing le 
entire determinant according to a well-lrnotra rule for the decomposition of determ.nar.ts as a lunctron of the 
determinantshelonging to these trvo rectangles taken separately. As to the first lemma, b, wason onhe cum - 
lant Iw Wo . . . Wi 1 Wi Wj-pi] being so representable, we know that when [wp w,y . . . . w, i wj , L*"! -i- ‘ 
aid .insUrL opposite signs. Suppose, now, that the theorern is true wlreu the nnmhe^r of 
elements in the tvpe does not exceed i ; then the roots of [w, w, . . . Wi_i], say of 4'i-i. l^eing ca , • « 
Id oTiw, aay of 40 being called 1, K-M. these may be arranged in the follotvmg order of 
