UNDER THE FORM OF A CONTINUED FRACTION. 
423 
Again, to form the system of which equation (12.) is the type, we write 
{boX-\-b^) .U — {a^x^-{-a^x^-\-a2^-\-a^y ={hf)X-\-b^)a ^ — (<*o'^*+®i-®^+«2‘*‘+^3)^2 
= - ff 0 • — «1 Kx"" H- (<^0 • «4 — «2 • ^ 2 )-^+ ( ^l«4 — ^2 • «3) • 
Combining the two equations of the first system with the first of the second system, 
we obtain the first simplified residue Ljj+L', where 
and 
0 
L'= b. 
bo 
b, 
^0 • ^2”i”^i • b\ 
bo 
b. 
aA+aA 
b. 
b. 
o^\ • ^2~l“®2 • b\ ~~ b^.d^ 
b, 
0 
a^A—bo-a^- 
By again combining the two equations of the first system with both of the second 
system, we have the determinant 
0 
ao.b, 
ci^.b^ 
bo 
b. 
b. 
b. 
®0* ^2~f” *^1 
U) . ^2 
«l-^2+«2-^l — ^1«3 
^2-^2 — bo.a^ 
b. 
0 
a^A—bo’Oi 
• ^2 ““ ®4 • bi 
which is the last simplified residue, or in other terms, the resultant to the system U, V. 
Art. ( 7 .). It is most important to observe that the Bezoutian matrix to two func- 
tions of the same degree (n) is a symmetrical matrix, the terms similarly disposed 
in respect to one of the diagonals being equal. 
Thus retaining the notation of art. (5.), so that 
(0, 1)=«|3 — bcc (1, 2) = hy — ca (2, 3) = cS — c?y 
(0, 2) = a 7 — ca (1, 3) = b^ — d(3 &c. 
(0, 3)=a^ — dec &c. 
&c. 
&c. &c., when n=\ the Bezoutian matrix consists of a single term (0, 1) ; 
when n~2, it becomes 
( 0 , 1 ) ( 0 , 2 ) 
( 0 , 2 ) ( 1 , 2 ); 
when n=i3, it becomes 
(0,1) (0,2) (0,3) 
/(O, 3)\ 
(0,2) -h (1,3) 
\(1, 2)/ 
(0,3) (1,3) (2,3); 
3 K 
MDCCCLIII. 
