OF '’n- UNDER THE FORM OF AN IMPROPER CONTINUED FRACTION. 477 
certain of the residues may be wanting, which will be the case when any of the 
secondary Bezoutics have their first or successive first terms affected with the coeffi- 
cient zero ; the equation connecting the residues with the Jacobian will then change 
its form (as some of the quantities Cj, C^, ... C„ will become zero) ; but I do not propose 
to enter for the present into the theory of these failing, or as they may more properly 
be termed, Singular cases in the theory of elimination. 
Art. (m.) The series last obtained for J (/, g) leads to a result of much interest in 
the theory, and of which great use is made in the concluding section of this memoir, 
viz. the identification of the Jacobian (abstraction made of the numerical factor n) 
with what the Bezoutiant becomes when in place of the n variables in it, Ui u^...Un, 
we write ... x, 1. Thus suppose / and g to be each of the third degree, 
and let 
Aj?^- 1 -Hcr-|-G 
B.r- 1 -F 
Fx+C 
be the three primary Bezoutics ; if we make 
x^=u x=v l=w, 
these may be written under the form 
AM-l-Hy-l-Gw=L 
Fiu-\- Bv-f Fip=M 
Gu-\- F?;-l-Cz^=N ; 
and if the Bezoutiant be called 9 , we have 
N=® 
au av aw 
The simplified residues to f and g are L, (L, M), (L, M, N), where (L, M) means 
the result of eliminating u between L and M, and (L, M, N) the result of eliminating 
u and V between L, M, N ; and by a theorem (virtually implied in the direct method* 
of reducing a quadratic function to the form of a sum of squares), if we call the 
leading coefficients of these quantities Cj, Cg, C 3 , we have 
(L, M)2 (L, M, N)2 ^ 
C 2 .C 3 - 
Hence when w=3 |.J(/ ^) = 9 when in 9, u, v, w are turned into x"^, x, I, and so 
in general for any values of n, the Bezoutiant correspondingly modified, becomes 
as was to be shown-f-. 
* Viz. that of M. Cauchy, adverted to in Section IV. art. 44-45. 
t Compare Jacobi, “ De Eliminatione,” § 2 . The general expression for the allotrious factor, I may here 
incidentally mention, is given under the head Theorem a, § 16, which comes quite at the end of the same paper. 
