430 
i\!R. SYLVESTER ON THE RESIDUES 
OF V- EXPANDED 
indefinitely extended ; — 
a I m p s 
I h n q t 
m n c r u\ (M-) 
p q r d V 
s t u V e 
r here begins with tiie value (1), and Do, Dj, Dg, D3, 
gression, 
1 ; a-. 
a I 
I r 
aim 
I b n \ 
m n c 
a I m p 
I h n q 
m n c r' 
p q r d 
D4, Dj will represent the pro- 
a I m p s 
I h n q t 
m n c 7 ' u . . (II.) 
p q r d V 
s t u V e 
so if we use the matrix 
a I m p s 
I' h n q t 
m n c r u 
p q r d V 
s t u V e 
the determinants Dj, Ds, D3, D4, D5 representing 
a ; 
a I 
I' b’ 
aim 
V bn,', 
m n c 
a 
I! 
m 
P 
I m 
b n 
n c 
q r 
will possess the property in question ; the line and column I, b; I’, b not being identical, 
the first determinant Do representing (1) must not be included in the progression. 
We shall have occasion to use this theorem as applicable to the case of a matrix 
symmetrical throughout, and we may term the progression (IT) above written a pro- 
gression of the successive principal determinants about the axis of symmetry of the 
square matrix (M), and so in general. Now it is obvious that the leading coefficients 
of the successive Bezoutian secondaries are the successive principal determinants 
about the axis of symmetry of the Bezoutian squares ; they will therefore have the 
property which has been demonstrated of such progressions ; to wit, if the first of 
them vanishes, the second will have a sign contrary to that of + 1 ; if the second 
vanishes, the third will have a sign contrary to that of the first, and so on. 
Art. (12.). Now let f.v and (px be any two algebraical functions of .r with the leading 
coefficients in each, for greater simplicity supposed positive : and in the course ol 
developing ^ under the form of an improper continued fraction by the common pro- 
cess of successive division, let any two consecutive residues (the word residue being 
