432 
EXPANDED 
MR. SYLVESTER ON THE RESIDUES OF ^ 
symmetrical matrix corresponding to this case will be the same as those formed from 
the mmmetrlcal matrix corresponding to /r and (where is <p(x) treated by 
aid of evanescent terms, as of the same degree as/r), with the exception merely of a 
constant multiplier (a power of the leading coefficient of /r) being introduced into 
each secondary. By aid of this observation, the proposition established for the case 
of two functions of the same degree may be readily seen to be capable of being 
extended, from the case of / and <p being of the equal dimensions in x, to the 
general case of their dimensions being any whatever. 
Art. (13.). Before closing this section, it may be well to call attention to the nature 
of the relation which connects the successive residues of fx and (px with these 
functions themselves, and with the improper continued fractional form into which 
^ is supposed to be developed in the process of obtaining these residues. 
If ^x be of n degrees, and/^ of n-\-e degrees in («), we shall have 
1 1 1 L, 
fx Qi— q3~ 9n 
where Q, may be supposed to be a function of ^ of the degree (e), and q,, q„ . . . q., 
are all linear functions of ^ ; the total number of the quotients Q^, q„ . . . ?„ being of 
course (n) when the process of continued division is supposed to be carried out unti 
the last residue is zero. Upon this supposition the last but one residue is a constant, 
the preceding one a function of ^ of the first degree, the one preceding that a function 
of X of the second degree, and so on. 
Let us call the residue of the degree s in .r, ; it will readily be seen that the 
successive complete residues arranged in an ascending order will be 
\,\.qn,'^Mn-l‘qn~^)’ 1 • ; ^CC., 
being in the ratios of the quantities 
1 5 qn ? 
Again, we shall have in general 
1 11 
1; 
— L,(p — 
(15.) 
A beinff an integral function of x of the degree n-/- 1, and L, an integral function 
of X of the degree (n+e )-.- 1 s and it is easy to see that th« successive convergents 
to the continued fraction — 
1 J_ J_ &c. 
O'!”" ?2 93 
have their respective numerators and denominators identical with those of the 
fractions 
A„_i 
LZ7’ k-, L,., • 
Adopting the language which I have frequently employed elsewhere, I call 3, a 
