VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 
535 
the disposition of the imaginary roots of equations appear to have led the way. 
Finally, the number of variables may be supposed to be arbitrarily increased, but 
made always inferior by a unit to the number of the functions in which they are 
contained, or which comes to the same thing, we may construct the theory of a 
system of homogeneous functions equal in number to the variables in them, which 
in its simplest case becomes the theory of Intercalations which has been here par- 
tially considered, and which (as has been shown) embraces (not as a particular case, 
but as an implied consequence and easily extricated result) the theorem of M. Sturm. 
London, June 25, 1853. 
General and Concluding Supplement. 
Art. (^?.). The expressions given in art. {n.) for the partial quotients of the con- 
tinued fraction represented by^, are restricted to the supposition of ail these partial 
quotients (except the first) being linear in x-, when the first partial quotient is linear 
the formula (B.) of that article continues applicable on replacing (D^ A 0 )by 1 . I was 
forcibly struck by the peculiarity of these formulae not ceasing to be true in conse- 
quence of the first partial quotient being supposed non-linear; and reflecting upon 
this, I was soon led to perceive that all the partial quotients might be supposed to be 
arbitrary integral functions of x, and the formulae would still continue to apply to 
any such of them as might happen to be linear, although, as it were, imbedded among 
a group of other non-linear partial quotients. From this it was but an easy step to 
perceive that the formulae A and B must admit of extension to the representation of 
partial quotients of any form, and that the dimorphism of the representation of the 
linear partial quotients could only be a consequence of the equation in integers u-\-v= 1 
having two solutions u=0, v= ] and u—l, y—0. I now proceed to enunciate the very 
remarkable general theorem (or as it may perhaps not inappropriately be termed 
Algebraical Porism), by virtue of which any partial quotient of a given degree in x 
belonging to an infinite continued fraction, all of whose partial quotients are alge- 
braical functions of x, may be expressed to a constant factor pres, by means of the 
numerator and denominator (or if we please either one of these) of the convergent 
immediately antecedent to and of the numerator and denominator of any convergent 
not antecedent to the partial quotient which is to be determined. 
Art. (l.). Theorem. Let Qj, Q^, ...Q^ — Q„, &c., each of an arbitrary degree in 
X, be then first partial quotients of an algebraical continued fraction; let be * 
the partial quotient to be determined and of the given degree let 
1 1 1 J 
Q, — Q2— 
and 
11111 1 ^{x) 
Qi- Qj- Q3- Q.i- F(^ ’ 
let u and v be any couple of integers of the couples which satisfy the equation 
u-\-v=Ui +^ ; then, as usual, denoting the product of the differences of each of one set 
MDCCCLIII. 4 A 
