500 MR. SYLVESTER ON A DEVELOPMENT OF THE MEIHOD OF ASSIGNING 
So that [?, for all values of ^ less than A will preserve an invariable sign, 
and consequently A is an inferior limit tofx. 
Art. (^.)* It may be remarked that the quantities 
1 1 .1 I ^ . t 
may be derived successively from one another, according to the same latv, from 
whichever end of the series we begin. 
If we take any two consecutive terms as 
the elfect of diminishing /a is to decrease the first of these two terms, and pro tanto, 
to tend to deduce the limit ; but on the other hand, being increased, there is 
brought into play an opposite tendency, which operates pro tanto to increase the 
value of the limit. _ 
Art (t ). It is of importance to remark, that by a right selection of the system of 
quantities p,, which enter into the composition of h h-.X, L may he made 
to coincide with the greatest root of [?. 9....?.] ; and so ,n like manner by a right 
selection of another system of these quantities, whereby to form A may 
he made to coincide with the least root of the same. Thus let p, he so 
chosen, that 7 n r —a 
= 0 q,-h=^0...q„-h=0 
are all satisfied by the same value of a:. 
Then 
exist simultaneously. 
Hence 
1 , J- _ 1 
92=p2+~ 93— 
1 1 
1 ^%—q^—q^ f ^3 — 93 93 q^- q^ 
1 1 1 
f^«-l = 9«-l“"^_2— qn-i Ql 
__i 
qn-2-‘"qi 
which is satisfied by making [9,1, 9 «-i, 9»-2---9i] , 1 j 
It remains then only to show that the greatest root of a’ m this equation substitute 
for ir in 9,, 9,, ... 9. will make all of one sign, and that the least root of .r 
similarly substituted, will also make them all of one, but a contrary sign, which n.aj 
be proved as follows. 
We have 
f..=9, ,..-[9o9j-?. fr,-.= [9.!=-?.-.]-k[?o?.-?-J‘ 
and by Lemma B the superior limit to [9, 9 i-?.] "’i" a superior limit also to 
9„ 9s9„-, 9.-.1 and to [9. <lc], [</< ?« -> 
