SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 501 
Consequently this superior limit will make have all the same sign as 
that of the coefficients of ^ in q„ q„ ... q^. And in like manner, the inferior limit to 
§'25 ••• 'tvill cause ^1, to have all the contrary sign to that of these 
coefficients. 
Thus then we see that when the coefficients of x in the partial quotients to — ex- 
foe 
pressed as an improper continued fraction form a single series of continuations of 
signs, by a right choice of the arbitrary constants (j,,, the superior or 
inferior limit given by this new method may severally and separately be made to 
coincide with the greatest and least real root, or each in turn with the sole real root 
ofy^r, if there be but one. 
Art. (^.). The general method of enclosing the roots of/r within limits is founded 
upon the combination of the two theorems above demonstrated. An arbitrary 
function <px one degree in x below, fx being assumed, and by aid of the auxiliary 
function (px^fx being thrown under the form 
C[q,, q^, ^..qi q\, q,, ...q\, q'[ {q\, {q\..., 
in which the coefficient of ^ is supposed to change sign in the passage from 9.. to q\, 
from 9;., to q'{, &c., a superior limit is found to each of the cumulants 
[9. ... [(9). (9)2. ..(?)«], 
taken separately, by means of the second theorem, and then by virtue of the first 
theorem the greatest of these superior limits is a superior limit to the cumulant 
bi 5^2. ■.9.- ...( 9 )i...( 9 )(o], 
and consequently tofx, and so mutatis rnutandis the least of the inferior limits of the 
same partial cumulants is an inferior limit to the total cumulant 
Art. (;?.). When all the roots of/r are real, if px be so assumed that all its roots are 
intercalated between those of/r, the partial quotients to ^ will form but one single 
series. In order that (px may fulfill this condition, it is necessary that the coeffi- 
cients ofpx shall be subject to certain conditions of inequality, not necessary here to 
be investigated ; but no conditions of equality, i. e. no equations between the coeffi- 
cients of px, are introduced by this condition ; or in other words, the coefficients* of 
px, the auxiliary function, are independent and arbitrary within limits ; and we have 
shown that in this case the auxiliary constants may be so determined that 
the limits may be made to come separately and respectively into contact with the two 
extreme roots. When all the roots of/r are not real, the quotients (however px is 
chosen) can no longer be made to form a single series. It still however remains true, 
that, by a due choice of the auxiliary function followed by a due choice ot the 
It need scarcely be stated that/'x is the simplest form of (px, -which satisfies the condition in question. 
