SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 49 
Art. (7.). The second theorem is the following: if q, ...q„, be linear functions of 
jc, say a^x-^b,, a^x-^h^...a^x-\-h^, in which the coefficients of x have all the same sign, 
and if we take the quantities ^2, . all having the same sign as tty a^...Un, but 
otherwise arbitrary, and make 
f^i r2 F'ft-2 
then the greatest of the quantities 
K~bi ^2 + ^2 ^n—in 
y ^ , , , j 
(I I an 
say L, is a superior limit, and the least of the quantities 
— —^2 — ^2 —kn — bn 
say A, is an inferior limit to the roots of fx. 
L and any value greater than L substituted for x will evidently make q,—h^-, 
92—^2', •••; qn—^n) all of them positive. 
Hence when x= or >L 5-1 is positive and >ja-i and 
1 7 1 ,11.. 
^2— e. IS positive, and 
positive, and >^„ 
^"2 
^■2 P’2 
and 
1 J. 1 _ 1 
5'a-2 pa-i Pn-l’ 
e. is positive. 
and consequently the cumulant \jiiq-2q2..^q,^, which 
remains of a constant sign when L and any quantity greater than L is substituted 
fora:. Hence L is a superior limit. In like manner A and any quantity less than 
A will evidently make f/i+A'i, q^-\-h-2-, ... 5'„+A-„ all of them negative, so that when 
x=. or <A ^r, is negative, and <—(^1 
1 7 1 . 
92— —<^2— IS negative, and < —u,^, 
'h P 
1 7 1 . 
93 — -<f^ 3 —- IS negative, and < — y.3, 
^2 P2 
and 
9 n- 
1 
^n-l — 5’n-2 
111. 
•- < IS negative. 
(/l pn — 1 pn— 1 
