SUPERIOR AND INFERIOR LIMITS TO THE REAL ROOTS OF AN EQUATION. 493 
L, Aa ; La Aa ; ... L„ A„ will be respectively superior and inferior limits to /x, (px and 
their successive residues. As a corollary, we see, of course, that L and A, the superior 
and inferior limit to the roots of the given function fx, must always lie between + oo 
and the greatest root, and between —co and the least root, of the arbitrarily assumed 
function px. 
Art. ( 53 .). Let us now assume somewhat more generally that px is any number 
of degrees in x lower than fx, which will cause the first quotient to be 
of the degree x and let us further suppose that px stands in such a relation 
tofx that the following quotients, qg^, qg ^, ... qg^, are of the degrees in x { 0 ^, 
being supposed not necessarily units, as they would generally be, but any positive 
integers whatever, as may happen in consequence of one or more of the leading 
coefficients in any residue vanishing), then 
£^_JL _J_ _J_ I J_ 
fa~qg^+ qg^+ 
where ^,+^2+^3+ and consequently /j? will be equal to the denominator of 
the last convergent above written, multiplied by a constant, so that we have now 
c.fx=^my.m^...mp, where 
t ^ I 1 
m{=-qg nu^QgA ...m=qg A . 
nd as in the case previously considered, so long as 
>1 
>2 
>2 
>1 
qe, or 
qe.. or 
qe. 
or , 
■■■qep or 
<-l 
<-2 
<-2 
<-l 
fx will have the 
same sign as qg 
Let now 
qB = 
1+ 
II 
=+< 
'^'••qep- 
= ±Cp, 
where 
Ci = 
1 C2=2.. 
•Cp- 
1 = 2 
Cp=l. 
Consider any pair of the above equations as ql.— cl= 0 . 
1st. Suppose all the roots of this equation are impossible, ql—c] must be positive 
for all values of x, and qg. can never lie between and -c^; moreover, since upon 
the hypothesis made, and qg.—Ci always retain the same sign, viz. that of the 
coefficient of the highest power of qg., it follows that qg. must also always retain the 
same sign ; for if we construct the two curves y=qgr\-CiW[id y—qg^ — c^, these will both 
lie on the same side of the axis of x, and never cut the axis, consequently the cuiwe 
which lies between them, must also lie on the same side as either of them, and 
never cut the axis. 
Hence, then, if the roots of the equation are all impossible, qg^ will always retain 
the same sign, and will never fall within the region bounded on its two sides by 
+ Ci and — C;. 
2nd. Suppose the equation to have one or more possible roots, and to the greatest, 
and \ the least (which of course, if there is but one possible root, will be identical). If 
