SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 505 
limit to «!, (^2, can make &> vanish unless each separate product in the expansion 
of a; in terms of iWj, •••*'« and the appurtenant apocopated cumulants vanish sepa- 
rately. 
Art. (X.). From the above theorem we may deduce the following law, viz. that if 
the roots of aii, a/2, be supposed to be arranged in order of magnitude, and X to 
be that one of them which is nearest to -f- 00 or to — 00, then if e is even it is im- 
possible for X to be a root of a/. Thus suppose e= 2 , and consequently a/=:a/^.a/2—^i-‘^2 1 
if X be a root of a/, and one of the two extremes of the roots of a/^, 0/2 put in order of 
magnitude, X cannot be a root of '<^25 for the roots of '0/2 are confined between the roots 
of a/2; but if X make a/ and a/i each vanish, we must have a/\.'a/ 2 ~ 0 , hence 0/1=0 as well 
as 4 / 1 = 0 , which is impossible. In like manner if a root of a/2 were the extreme root, 
the same impossibility could be in like manner established. 
Again, suppose e= 4 , so that 
1 2 
CD' ,*(JD 
3 
1 
CO' %'Oi' m'O} 
1 2 * 3 _|. 
CO', •'UO^tCO' ,'(/} 
1 *^2 3 4 1 
»2-"3 
a/2.«/3.«/4 
CO 
l-“2- 
"3-W4 J 
Let X continue to denote one or the other extreme of the roots of 4/, 0/2 4/3 4/4. We 
must in each case, if X makes 4;=0, have 
a/i.a/ 2 .a/ 3 .a/_^ = 0 ; aj[,'a/ 2 .a/ 2 .a/^ = 0 ; a/^.a/ 2 .'a/ 3 .a /^=0 ; 4/, .4/2. 4/^. '4/4 = 0 ; 
4 /'i .' 4 /' 2 .' 4/3 = 0 ; 4 /' i .' 4 / 2 . 4 /^.' 4/4 = 0 ; 4 /i . 4 /^ . ' 4 /^ . '4/4 = 0 ; 4 /'i .' 4 /^ .' 4 /^ . '4/4= 0. 
Now suppose that X is a root of 4/,, then the equations remaining to be satisfied are 
i/'l.'4/2. 4/3, 4/4 — 0 ; 4/'i.'4/^.'4/3 = 0 ; 4/'i.'4/2.4/^.'4/4 = 0 ; 4 /'i . ' 4 /^ . ' 4 /^ . '4/4 = 0 . 
Since a/, and a/\ cannot both be zero together, X cannot make a/[ or 'a/, zero ; and be- 
cause X is an extreme to the roots of a/2, -/g, a/^, X cannot make 4/^ or 'a/2 or 4/3 or '4/3 or 4/4 
or '4/4 zero, so that in fact when x=X none of the singly accented quantities 4/ can 
be zero. As regards the doubly accented quantities a/, the same thing cannot be 
affirmed, because if any Q contains only one element the corresponding value of 4/ 
with a double accent vanishes spontaneously. Again, any of the unaccented quanti- 
ties 4/ may vanish, because we may suppose any of these to have an extreme root X. 
Consequently the first, second and fourth of the equations remaining to be satisfied, 
might be satisfied on making the necessary suppositions as to the form of the quan- 
tities 4/ and the values of the extreme roots; but the third remaining equation 
4/, .'4/2. 4/3. '4/4=0, in which only singly accented quantities 4/ occur, remains incapable 
of being satisfied on any supposition whatever. And the same thing would be true 
if we suppose X to be a root of any other 4/ instead of a/^. Hence X cannot make a /=0 
when e= 4 . 
In like manner, if e be any even number 2?, there will be an equation 
4/'i . '4/2 . aj'2 . '4/4 . 4/: . '4/g . . . 4/2,_ 1 . '4/2e = 0 
to be satisfied by that value (if it exist) of x which, besides being an extreme (on 
either side) of the roots of 4/,, a/2, ... 4/2* arranged in order of magnitude, also makes 
4/=0. But as such equation cannot be satisfied, neither extreme root of the roots of 
3 u 2 
