SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 497 
Lemma A. The roots of the ciimulant which each element is a linear 
function of x, and wherein the coefficient of x for each element has the like sign, 
are all real, and between every two of such roots is contained a root of the cumulant 
and ex converso b. root of the cumulant [?2 ^3...?,], and (as an evident 
corollary) for all values of § and / intermediate between 1 and i the greatest root of 
[^1 ?2.-.?i-i^,] will be greater, and the least root of the same will be less than the 
greatest and least roots respectively of [5'p g'p+i...g'p'_i ^p/]. 
Lemma B. For all values of the elements g'l g'2...g'„, the cumulant 
^0. 5'm+I ^<o + 2'*-5'J 
= [^1^2... ^<0-1 X [^<. + 1?. + 2 ... 9 „] 
— [91 ^2 • . . S'o.-l] X [^,+2 . . . 9 J . 
Thus ex. gr. the cumulant [ahcd], i. e. ahcd—ab—cd—ad+l, 
= [_al'\ X [c</] — [a] X [c?] = (a& — 1 ) {cd— \) — ad, 
and [ahcde']^ i. e. abode— ahc—ahe—ade—cde-\-a-{-c-\-e=:.[ahc'\\ae~\~[ab']\e], 
i. e.= {abc — a — c){de—l') — (ab—})e. 
Art. (( 3 .). Now suppose that qiq^.-.q^ q^+^.-.q^'dre all linear functions of .r, and that the 
coefficients of x have all one (say the positive) sign inq^q^... q^, and all the contrary signs 
in q^+i-.-qn, and let L be not less than the greatest root of [g'ig2---9j or of 
and also let A be not greater than the least root of each of these same two cumulants ; 
then by lemma A, L and A will also be respectively greater than the greatest, and 
less than the least roots of [g'l g'2-''5'w-i3 nnd of [g'„+2«--g'J. Now the coefficient of 
the highest power of x in both [q^q^...qj] and in [qi q^-.-q^-i] is positive, but as to 
[g'„+i...9„] and [^„+2...g'J is of contrary signs in the two, viz. negative in that one 
of those cumulants which contains an odd, and positive in that one of the two which 
contains an even number of elements. Hence by virtue of Lemma B, L and any 
quantity greater than L substituted for x will make [g'l g'a.-.g'J to have always the 
same sign, and in like manner it may be shown that A and any quantity less than A 
substituted for x will also cause {jliq^.-.q^ to retain always the same sign. Hence 
L and A are superior and inferior limits to [g-, q^.-.q^ ; and the same reasoning would 
magnitude ^3. . . Aj_i hi-i ki ; and if the roots of [wj Wj Wj+i], say of 4'i+i, be called 4 k+i, 
from the fact of the leading coefficients in and 4 'i+i expanded according to the powers of x having the 
same sign, it follows that when <r=co, and ^i+i have the same sign, but they have contrary signs 
when X k ; but 4 'i—i does not change its sign between a; = oo and x—k, hence 'pi+i does change its sign be- 
tween X CO and x k^, and therefore a root of 4 'i+i lies between 00 and k ^ ; in like manner precisely it may 
be shown that a root of xj/i+i lies between — co and 4; and since changes its sign between 4 and 4^, 
between k,^ and — k, and between 4_i and 4. 'I'i+i must likewise change its sign between one and the other 
extremity of each of these intervals, and hence the roots 4 4 4 +i are intercalated between 00 , k,^, 4, 
— GO , or which is the same thing, 4, 4 > — h are respectively intercalated between 4, h’---h+i ; consequently, 
if the theorem is true up to i, it is true for z-j- 1 , and therefore true universally ; but is manifestly true when 
i— 2, for then a;= + co makes [wj, wj, f. e, Wj positive ; but tt;i = 0 makes it negative, which proves the 
theorem contained in Lemma A. 
3 T 2 
