540 
MR. SYLVESTER ON THE CONVERSION OF A SERIES 
types, the point for each fracture of the total type being marked by a change of sign 
in the elements of the type for the value easily seen therefore fiom this, 
that if ^ is the generatrix of the cumulantin question, the number of such fractures 
(/. e. the number one less than the number of partial cumiilants) will be the number 
of changes of algebraical sign in the signaletic series, consisting of the leading coeffi- 
cients in and in each of the odd-placed complete residues respectively^, together 
with the number of changes of sign in the signaletic series, consisting of the leading 
coefficients in and in each of the even- placed complete residues respectively. 
The syzygetic theory of two algebraical functions, and the allied theoiy of alge- 
braical continued fractions with their principal applications, may, I think, now be said 
to be completely made out, as well for the singular cases as for the general h\ po- 
thesis. 
Art. (k). I will conclude with observing that the theory within developed gives the 
means of transforming (explicitly and without the aid of symmetrical functions) into 
an algebraical continued fraction, any given sum of algebraical fractions of the form 
^l_ ■ ^2 I ^3 I 
x—h,p x~h^ ' x—h^ 
where each c and h are supposed known. For let the above sum be called then 
if hg, Cg be used to denote any pair of corresponding terms of the h series and the c 
series, we have as is well known and easily proved. Again, if represent 
the simplified denominator of the ^th convergent to the continued fraction equal to 
which is to be found, say 
(Jb X) 
we have ^ {^x—h^{x hn), 
hi+i hi + 2 
^(Ai h2 ...hi)^hi — /q) (x — A 2 ) . . . (x — hi) 
Therefore 
fhi.fh2...fhi 
= { 2(c2C2 . . . Ci+ 1 ) • • • KA } ; 
and the simplified (i-l-l)th quotient, i. e. the value of Ai+iX-j-Bi+i, when divested of 
the allotrious factor, has been proved to be equal to 
