SUPERIOR AND INFERIOR LIMITS TO THE REAL ROOTS OF AN EQUATION. 491 
where it will be observed that the excluded region lies between +2 and —2 for all 
the intermediate quotients, but between only + 1 and — 1 for the first and last quo- 
tient. Then is positively or negatively greater than 1 , therefore — is a positive or 
negative fraction, but is positively or negatively greater than 2 ; therefore will be 
of the same sign in q^, and also will be positively or negatively greater than 1 ; 
therefore ^ will be a positive or negative fraction, but q^ is positively or negatively 
greater than 2 ; therefore will be of the same sign as q^, and also will be 
positively or negatively greater than 1 ; and proceeding in this way, we find that all 
values of from 2=1 to i=in~\, will be of the same sign as qi, and positively or 
negatively greater than 1 . Finally, ~ will be a fraction, and therefore, since a is 
positively or negatively greater than 1 , will have the same sign as {q„) 
(but of course is not necessarily greater than 1 , nor would that condition serve any 
purpose were it satisfied). We infer consequently, that when the conditions {a) are 
satisfied, yj^, yj^, ^ 3 ...^^ will respectively have the same signs as q^ ^ 2 *** 9 ^re 5 and there- 
fore D=yji.yj 2 .y^ 3 ...y„ has the same sign as qi.q^^q^.^.qn- Now suppose 
q, = a,x+b, q<i=a2.x-\-h2...q^-=a^.x-^b^, 
and solve the equations 
a,x-\-b,= ->rc, = - 1 -^ 3 ... r-l- 6 „_,= c„_, a^.x-{-b^= 
a,.r-l- 6 ,= -c, « 2 ‘ 3 ^+^ 2 = — C 2 ...ff„_,..r-l-^„_,= — c„_, a^.x-\-b„——c^, 
where c,= l 0^=2 ^ 3=2 c„_i = 2 c„=l. 
Whenever in any one of the n pairs of equations above written the coefficient of x is 
positive, the upper equation of the pair will bring out the greater value of:r; but 
when the coefficient is negative the lower equation will give the greater value. 
Take the pair 
a^x-^-bi — Ci 
aiX-\-b(= — Ci. 
If a; is positive a^x+b^ will always be positive, and greater than c, between a^=co and 
jr= the greater of the two values of ; if a. is negative a,x-\-b,w\\\ always be negative, 
and less (i. e. nearer to co ) than — for all values of x between the same limits as 
befoie. So again it will be seen in like manner, that whether a,- be positive or negative 
between j:=— co and x= the lesser ol the two values of x corresponding to the 
above pair of equations, ayc-\-bi will always retain the same sign, and will be greater 
than +c,-, or less than — Cj, according as is negative or positive. If, then, we take the 
greatest of the greaters of the n pairs of values of x, i. e. the absolute greatest of the 
2n values, and the least of the lessers, i. e. the absolute least of the same, say L and A 
between L and A, q^, q^, ...q^ will each always retain an invariable sign, and will then 
fall without the limits +Ci +C 2 ie„_i so that between -l-oo and L 
and between A and— co, yj^.yj^...yj^^ i,e. a constant multiple of f{x), will retain the 
