498 Ma. SYLVESTER ON A DEVELOPMENT OP THE METHOD OF ASSIONING 
evidently apply if we had supposed the signs of the coefficients of x in the first partial 
series of elements to have been negative, and in the other series of elements to have 
been positive. ^ r .u 
The greatest and least roots of X ^ 
dition to which L and A are subject, and may be taken in place of L and A 
respectively. They will accordingly be superior and inferior limits to the cumiilan 
[^1 5'io+i"'9«l‘ 
Again, by virtue of theorem (B.) it may readily be shown that 
[g'l S'ioi+1 ?<02+2-'-M“2+1'"9»] 
= [^1 X [</'on + l X [9al2+l---9»] 
— X [9oi.+ 2---5'J X [gai,+ l-"9«] 
— X X [^lo^+a- ••</«] 
and hence if q. q,...qn are all linear functions of ^ in which the coefficients of x have 
all the same algebraical sign in any one (taken per se) of the three series 
S'! ^i2---9io, ’ 5'iOi+l”*?a,, ; ■’ 
but so that this sign changes in passing from one series to another, it is easily seen, 
by the same reasoning as in the preceding case, that the two positive and two nega- 
tive products on the right-hand side of the equation all give the same sign to the co- 
efficient of the highest power of and consequently that if L and A be superior an 
inferior limits to 
and consequently by Lemma A, to 
[^1 ^2 •••?«.-!] 5 [S'lPi + a-'-S'Jj + [5'lUi + 2---?'al,-l]5 aildtO [9„„+2 ?n]. 
Lor A substituted for x will cause \_q,q....q,:\ to retain always the same sign, and 
will consequently be superior and inferior limits thereto ; and so in general ; w lence 
it follows, returning to the theorem to be demonstrated, that the greatest and east 
roots of P -1 
[?! 92- --^i] X [^/i+i 5'i+2---5'd X ... X 
will be superior and inferior limits to the cumulant [<?, i/o . I e. to C./x*, and 
therefore to fx, as was to be proved. 
If if expanded as a continued fraction by means of the common measure process gives rise to the quo- 
tients gC? 2 ..- 5 «. and if L,, W.-.U-,, L„ be the leading coefficients of the successive simplified residues 
(L„ being, in fact, the final simplified residue, i. e. the resultant to (px.fx), we must have C[ji. •• •?« 
fx—C\_qi, ? 2 .. .?„]i where (supposing (px to be of « — 1, and/if of n dimensions in x), 
LL 2 • K-4 
