494 
MR. SYLVESTER ON A NEW METHOD OF FINDING 
the leading coefficient of qi is positive, the greatest root (/) of the equation qi c— 0 will 
exceed the greatest root of (/') of the equation qer\-Ci=-0-, for between j7=cc and x—P, q^. 
must go through all values intermediate between oc and — q ; hence there must be a 
quality/ intermediate between I’ and -{-co , which will make q^^c^. In like manner, if the 
leading coefficient of q,.\% negative, it will be seen that the greatest root of ^,.+c.-=0 
will exceed that of q,.— c^~^. Moreover, in the one case q, will be always positive 
and greater than q, and in the other always negative, and less than q. In every case, 
therefore, between +oo and /;, q^^ retains the same sign, and does not fall within the 
region bounded by +Cj and — the same thing may be shown to be true for all 
values of x between -oo and Hence, then, by the same reasoning as that employed 
in the preceding article, we are enabled to affirm, that if we form the equation 
- 4 )( 9 ^«~ 1 ) = 0 , 
its greatest root will be a superior limit, and its least root an inferior limit to the 
roots of the equation /i=0, whatever be the value of the assumed function (px ; and if 
the above equation (4'*) 1^^® real root, all the roots oifx will be imaginarjr. 
Art. (54.). In the preceding two articles it has been supposed that all the quotients 
are taken integral functions of ^ ; but the process of successive division may be so 
conducted as to give rise to quotients of the form 
Suppose then that we have in general 
fx ^1+ ^2+ 
where q„ q^, ... q^ are each of the general form above written (but of course / and / 
being not necessarily the same for any two of the quotients), and suppose that the 
sum of the degrees in x of q^, q^, ... qu, is where t is essentially (as it must be) 
positive. Then we shall find, as in the last article, that L and A being called the 
greatest and least roots of {q\-\){ql-4.) ...{ql_-A){ql-\), D the denominator of the 
last convergent to the continued fraction above written, will never change its sign 
between + oo and L, nor between A and — oo ; but here we shall have 
fx—¥..x*xT). 
Hence x\D will be invariable in sign within each of these two intervals. 
1st. Let t be even; then /(x) will be invariable in sign, whatever L and A maybe 
for each such interval. 
2nd. Let t be odd ; then if L is >0 and A<0,/(.r) cannot change its sign in either 
interval; but if L is <0 or K>Q, fx will change its sign as x passes through zero, 
but will be invariable for each of the three regions contained between +oo and L, 
L and 0, or 0 and A (as the case may be), and A and — oo ; so that universally L and 
A will be a superior and inferior limit to the roots of fx, making abstraction of the 
roots (if any such there be in fx) whose value is zero. 
Art. (55.). I shall close this section with oflfering (for what it is worth) a bare 
