UNDER THE FORM OF A CONTINUED FRACTION. 
417 
Art. (3.). This may be done by the following method, which is extremely simple, 
and would admit of a considerable extension in its applications, were it not beside 
my immediate purpose to digress from the objects set out in the title to the memoir, 
by entering upon an investigation of the special or singular cases which may arise in 
the process of forming the continued fraction, when one or more of the leading 
coefficients in any of the residues vanish ; such an inquiry would require a more 
general character to be imparted to the values of the quotients and residues than I 
shall for my present purposes care to suppose. 
Let us begin with supposing €=1, and write 
= Si.C. J ' 
Let %// be the fiist residue ^nd of and therefore of so that a; is the second 
residue of -• 
9 
Let <y=/v(^), 6; being entirely integer, and X a function of the coefficients in/and <p. 
If we make N and D being integer functions, D will evidently be L" ; where L 
denotes the first coefficient in the simplified residue and is evidently of two 
dimensions in a, (3, &c., and of one in a, b, &c. ; Ba is therefore of 2 X 2-{- 1, L e. five 
dimensions in a, /3, &c., and of two dimensions in a, b, &c. ; but u (by virtue of what 
has been observed of the equations in system (5.)) is of three dimensions in a, j(3, &c., 
and of two m a, b, &c. Hence N is of two dimensions in a, /3, &c., and of none in 
a, b, &c. This enables us at once to perceive that N=a^, 
for -ip is of the form J‘—(px-\-q)(p, | 
and u is of the form J 
But N=0 makes a vanish, and therefore, upon this supposition,/" and (p would appear 
to have a common algebraical factor that is to say, N vanishing, would appear to 
imply that the resultant of / and p must vanish, so that N would appear to be con- 
tained as a factor in this general resultant, which latter is, however, clearly inde- 
composable into factors— a seeming paradox— the solution of which must be sought 
for in the fact, that the equation N=:0 is incompatible with the existence of the usual 
equations (7.) connecting/, p, ^ and but this failure of the existence of the 
equations {, .) (bearing in mind that N has been shown to be a function only of the 
set of coefficients a, f5, &c.), can only happen by reason of a vanishing whenever N 
vanishes ; a must therefore be a root of N, or which is the same thing, N a power of 
(a) and hence N=a^. 
The same result may be obtained a by actually performing the successive 
divisions ; if the coefficients of any dividend be a, b, c, d, &c., and of the divisor 
3 I 2 
