538 
MR. SYLVESTER ON AN UNLIMITED ARBITRARY ALGEBRAICAL 
Art. (1.). When 0(a^) = F'(a^), 
becomes identical with 
...a 
hi h^ ■ 
_Ai+„^+i A2+„.^g..-A„ _ 
and we may consequently (using an extreme term in tlie forms in tlie polymorphic 
scale of forms representing Qi+i), write 
= 2^(A^A2...A<.;^J(/^A0Vi^2)'•••(/^^-.•+l)'('^“^‘)(■^“^^)•••(•^~^■ 
Art. (?.). The following observations will serve to complete the theory of the 
singular eases in the expansion of an algebraical continued fraction. 
Preserving the notation of art. (^.)3 let 
Then (calling the roots of Fa;, A. h...h.) the (i)th simplified residue to m accord- 
ance with the general formute for the residues in the second section (for greater 
simplicity selecting an extreme term of the poly.norphic scale), will be represented by 
OA, <l>A.<I»A....<f(A. Ji _ /, ) (,r- A,) (i- A.) . . . .(a; - A,,) , 
Ai A2 A3 ...A 
j_Aj+jrj Ag+s’j 
which will be of the form &c, all tile terms containing powers of a; superior 
to a. vanishing by the coefficients becoming zero. If in the above expression we 
should use ir. in lien of where is <r, diminished by any integer inferior to we sho.i d 
get other forms of the same residue, but these will all be of higher dimensions in the 
roots or coefficients than the one just given, and in fact the forms thus obtainei 
corresponding to the values v„ ..-1, ..-2 a,-»,+ l substituted for a. m siicces^ 
Sion would by aid of the relations of condition between the coefficients of Oa and 
Fa implied in the value of <», admit of being exhibited as a scale in which each form 
would be an exact algebraical product of the form which precedes it, multiplied by a 
function of the coefficients, and did space permit thereof it would be perfectly easy to 
give the forms of these multiplicators. But I pass on to the representation ot what 
fs more material, viz. the form of the complete residue in the case supposed, merely 
observing (as an obiter dictum) that the existence of each singular partial quotient 
meaning thereby a quotient non-linear in x) only affects the form of the sing e 
limplified residue in immediate connexion with itself, and not at all the form of the 
Other residues antecedent or subsequent to that one. 
Art (n ) Let the ith simplified residue be called R, and the corresponding coiii- 
plete residue [R.], then applyinga method similar to the method given in Section I., 
we shall find that 
