fx 
Jx the form of an improper continued fraction. 479 
as it may readily be shown to be. And the formula (A.) may be verified in precisely 
the same manner. There is no difficulty in finding the values of C and C', which are 
products of powers, some positive and some negative, of the leading coefficients in 
the simplified residues, and recognising that they satisfy the equation (E.) ; when (px 
is of one degree below/r this equation is of the form C+C'=0. 
Art. (o.) When px—fx, this expression for the (i+l )th simplified quotient becomes 
2(Di A), as previously found; the correlative expression will be 
k being any root of which is equal to the former expression. The general 
expressions above given for the simplified quantities are of course integral functions 
of A and A, although given under the form of the sums of fractions, by virtue of the 
z\ 
well-known theorem that where ^ is an integral function of A, and the summa- 
tion comprises all the roots (A) of fh —0 is always integral. 
Art. (/).) It will be found that for all values of ^ greater than unity 
and that 
Ihe theorem of art (ri.) is in effect a theorem of Cumulants of the form 
[Q,(a:), q,{x), ...qt{x) ...q,{x)'\, 
where the elements are all independent of one another, and where /r represents 
[Qi(.r) q^ix) q^{x)...q^{x)~\ and (px represents \jq^x, q^ix), .,.q^{x)], 
n being any number whatever greater than i ; this makes the theorem still more 
remarkable. 1 he urgency of the press precludes my investigating for the present the 
moie general theorem which must be presumed to exist, whereby qi+^ can be con- 
nected with lq,q,q,...q;], or [<?, <73 • • • 9 J , and with Iq^q^q.-.-q^^,'] and [_q,q,...qi^,'\, 
when each {q) represents a function of an arbitrary degree in .r. The theorem so 
generalized would comprehend the complete theory of the quotients arising from the 
process of continued division, without exclusion of the singular cases (at present 
supposed to be excluded) where one or several consecutive principal coefficients in 
one or more of the residues, vanish. 
Art. (5-.) The complete statement of two twin theorems suggested by and intimately 
connected with the biforrn representation of the quotients given in the preceding 
article, is too remarkable to be omitted. 
Suppose px=f'x, and let the successive convergents to 4 - be called 
jX 
where the subscrolet index to t or T indicates the degree in x. Then if we call the 
MDCCCLIII. 3 
