EXPRESSED IN TERMS OF THE ROOTS OF (f)X, fx. 
447 
(by reason of the equations ^^=^2 ^3=% with ( — )^X the second factor 
of the denominator, and the second factor of the numerator with ( — )®X the first 
factor of the denominator; hence the coefficient of in 3 is —4. 
In like manner the only effective terms of 3^2, 2 will be 
J 
X 
nz 
??4 
1_^^3 
Uh 
h h 
hehr^ 
r/q 
A, h 
5 he 
hP 
X 
% ^4 
Jh 
h 
_ni ??2_ 
% 
X 
~^2 
?74 
[_ 
Ih, 
A 3 A 5 
he h._ 
[/h 
kj 
0 he 
hr 
n -2 ''li 
K 
X 
X 
^2 ^3 
■ 
^3 
Jh 
hji. 
hek_ 
r 
hjl 
5 he 
hj 
X 
[_ a 2 
hs 
_^2 ^3_ 
% ^4 
X 
nx 
^2 
A, K 
A3 
hJh 
Ag hy _ 
r A3 A, A. he 
A;^ 
X 
''ll ^2 
_ A2 
_^3 ^4_ 
^2 ^4 
X 
^1 ^3 
A; A3 
_ A2 
hi As 
Ag h^_ 
A, A4 A., he 
A?^ 
X 
% 
Ji A3 
_n-2 'ii_ 
^2 ^3 
X 
^54 
~ 
JiJh 
Jh 
A3 As 
he hj_ 
A2 A4 As he 
A;] 
^■2 ^3 
Ifi^ 
X 
_?Ji '^i_ 
Any other term will necessarily contain in the numerator a factor, whose symbolical 
representation will contain one of the quantities jji '/j.2 ^3 Vi in the upper line, and one 
of the quantities /q, having tlie same subscript index in the lower line, and 
which will therefore vanish ; the number of effective terms being evidently the 
number of ways in which four things can be combined 2 and 2 together, and the 
value of each term is evidently ( — )^'^.( — 1)^M, so that the entire value of the 
coefficient of n\-nl in ^^2,2 is +6. 
Pi •ecisely in the same rnannei’, M^e shall find that the leading coefficient in ^ will 
contain the term the ( — 1) resulting from the operation (— l)* k(— 1)^-^, 
and in the term the +1 resulting from the operation (— \ Hence it 
appears that 3^,, 4; 3; ^2,2; ^3,1; ^4,0 are to one another in the ratios of 1 ; —4; 
6 ; — 4 ; 1 ; and so in general for any values of m, rt, i {i being less than m and less 
than n) it will be found that 
i-2? • • • *^i, 
will be in 
the ratios of the numbers 
i—l 
_ j ^3(m-3) 
1 ; 
(-ir-.H ; 
(- 
i, 0 
2 — 1 2—2 
(-lyM 
Art. (27.)- The method employed in the preceding investigation will enable us to 
affix the proper sign and numerical factor to ^0,1 of ^1,0? or in general to in 
order that it may represent the Bezoutian secondary of the degree i in x. [This 
latter has been already identified with the simplified residue obtained by expanding 
under the form of an improper continued fraction.] For this purpose, it will be 
sufficient to compare a single term of any such ^ with the corresponding one in the 
Symmorphic Bezoutian secondary. Let us first suppose that m = /'and (p being of 
MDCCCLIII. 3 N 
