RELATIONS OF TWO ALGEBRAICAL FUNCTIONS. 
415 
Section I. 
On the complete and simplified residues generated in the process of developing under 
the form of a continued fraction, an ordinary rational algebraical fraction. 
Art. ( 1 .). Let P and Q be two rational integral functions of .r, and suppose that the 
process of continued successive division leads to the equations 
P -MoQ +R,=0' 
Q — M,R,+R2=0 
Rj — JVl2R2~}~R3— 0 !>■ 
( 1 -) 
so that 
1 1 1 
P~Mo— Ml— M 2 — 
&c 
( 2 .) 
which is what I propose to call an improper continued fraction, differing from a 
proper only in the circumstance of the successive terms being connected by negative 
instead of positive signs. 
Mo, Ml, M 2 , &c., Ri, R 2 , R 3 , &c. are, of course, functions of a? : the latter we may 
agree to call the 1st, 2nd, 3rd, &c. residues (in order to avoid the use of the longer 
term “residues with the signs changed”) ; and by way of distinction from what they 
become when certain factors are rejected, we may call R„ R 2 , R 3 , &c. the complete 
residues. Each such complete residue will in general be of the form N, and D, 
being integral functions of the coefficients only of P and Q, bui an integral 
function of these coefficients, and of a;: p, may then be termed the ith simplified 
N 
residue, and ^ the /th allotrious factor. Suppose P to be of m and Q of w dimensions 
in J 7 , and m—n=e, the process of continued division may be so conducted, that ail 
the residues may contain only integer powers of x ; and we may upon this supposition 
make M, of e dimensions, and M„ M 2 , M 3 , &c. each of one dimension only in so 
that Ri, R 2 , R 3 , . . . . will be respectively of (n—\), (n—2), (n—3), &c. dimensions 
in X. 
P and Q are supposed to be perfectly unrelated, and each the most general function 
that can be formed of the same degree. From ( 1 .) we obtain 
Ri = Mo.Q-P 
R2=MiRi — Q 
= (MoMi-l)Q-Mi.P 
R3=(M„MiM2+Mo+M2)Q-(MiM2-1)P 
&c. = &c. 
3 I 
(3.) 
MDCCCLIII. 
