506 MR. SYLVESTER ON A DEVELOPMENT OF THE METHOD OF ASSIGNING 
can be a root of as was to be proved. Consequently, unless cx is so 
assumed that the number of changes of sign in the coefficients of a- in the quotients 
resulting from ^ expanded as an improper continued fraction is even (for if the 
changes from sequence to sequence are odd the number of sequences themselves is 
even), the method of limitation in the text cannot give the means of drawing either 
limit indefinitely near to one or the other extreme roots oifx. 
Art. (fA.). It now remains to prove the converse, and to show, 1st, that when the 
number of changes is even, I e. the number of sequences odd, this coincidence can 
always be effected; and 2ndly, that it is always possible when/r has one or more 
real roots, so to assume (px that the number of sequences shall be odd. 
The first part of the proposition is easily proved. Thus suppose e=3, so that 
. 0^2 . iyj — . 'iya • "3 — • "2 • '<^3 + • '"A '"3- 
If we suppose X either extreme of the scale formed by writing in order of magni- 
tude, the roots of ^y,, ^^3 to be a root common to a., and to ..3 and which 
last equation may be satisfied by supposing the type to consist of a single element, 
the separate equations 
l>Jl = (y'i.'a^2.&»3=0 CO^.a'. 2 .'CiJ 3 = 0 = ^ 
will all be satisfied ; and so in general it may be shown without difficulty that if 
e" 2 s-\-l, and if X be a root common to &)i = 0 cu^^O cu^=0...a}.2t+i = ^> and if ^y^, 
be all simple linear functions of so that consequently ^2=0 'co ',=0 ...'co-2. = Q, each 
separate term in the development of . will vanish singly and separately, and conse- 
quently A will be a root of ^ : for since X makes ^^=0 0^3=0. every product 
in the developed form ^y, in which a;,, do not each bear at least one accent 
will vanish ; and if we consider any product in which 0;^, <v3---<^2.+i are all accented, 1 
in any two of these immediately following one after the other as a.u-i, ^2a-+i, an accent 
falls to the right of the first, and to the left of the second, the intervening term .^2.- 
will bear a double accent, and will therefore vanish, since 0-2, is supposed to be a 
linear function of .r ; but it is impossible when every a; is accented to prevent two 
accents of contiguous odd terms in any such product, from falling to the right ot t le 
left and to the left of the right, term of the two, since the contrary would imply that 
all ’the accents would fall to the right, or all to the left, which, as above remarked, 
is impossible, on accoun of the two extreme terms being only simply accentable, i. e. 
c, only to the right, and only to the left. Hence, when a’ substituted tor X makes 
0), C02...C02.,+1 all vanish, and when 0-4, ... <^^2* are all linear functions ot x, x—X w i e a 
root of . . 1 • . . ..cLt 
Art (t) I believe tliat the remaining part of the proposition may be rigoio 
demonstrated, viz. that when any of the roots of. /x are real, and the number of odd 
integers not exceeding the index of the degree of/r is m, and the number of ,n, agi- 
nary pairs of roots in/x is g., fx may be so assumed that the quotients to expanded 
