RELATIONS OF TWO ALGEBRAICAL FUNCTIONS. 
409 
that of the simplified residues, generated by a process corresponding to the develop- 
ment of ^ under the form of an improper continued fraction (where the negative 
sign takes the place of the positive sign which connects the several terms of an ordi- 
nary continual function). As the simplified residue is obtained by driving out an 
allotrious factor, the signs of the former will of course be governed by the signs 
accorded by previous convention to the latter; the convention made is, that the 
allotrious factors shall be taken with a sign which renders them always essentially 
positive when the coeflficients of the given functions are real. I close the section 
with remarking the relation of the syzygetic factors and the residues to the con- 
vergents of the continued fraction which expresses and of the continued fraction 
which is formed by reversing the order of the quotients in the first named fraction. 
In the second section I proceed to express the residues and syzygetic multipliers 
in terms of the roots and factors of the given functions ; the method becoming as it 
may be said endoscopic instead of being exoscopic^, as in the first section. I begin in 
arts. (14.) and (15.) with obtaining in this way, under the form of a sum or double 
sum of terms involving factors and roots of f and <p, and certain arbitrary functions 
of the roots in each term, a general representative, or to speak more precisely, a 
group of general representatives for a conjunctive of any given degree in x to/and (p, 
i. e. a rational integral function of x, which is the sum of the products of f and (p 
multiplied respectively by rational integral functions of x, so as to vanish of necessity 
when f and <p simultaneously vanish. This variety of representatives refers not 
merely to the appearance of arbitrary functions, but to an essential and precedent 
difference of representation quite irrespective of such arbitrariness. 
In articles (16.), (I/-)? (18.), (19.), (20.), (21.), I show how the arbitrary form of 
function entering into the several terms of any one (at pleasure) of the formulee that 
represent a conjunctive of any given degree may be assigned, so as to make such 
conjunctive identical in form with a simplified residue of the same degree. The form 
of arbitrary function so assigned, it may be noticed, is a fractional function of the 
roots, so that the expression becomes a sum or double sum of fractions. I first prove 
in arts. (16.), ( 17 .) that such sum is essentially integral, and I determine the weight of 
its leading coefiicient in respect of the roots of/* and pi (this weight being measured 
* These words admit of an extensive and important application in analysis. Thus the methods for resolving 
an equation (or to speak more accurately, for making one equation depend upon another of a simpler form) 
furnished by Tschirnhausen and Mr. Jerrard (although not so presented by the latter) are essentially 
exoscopic; on the other hand, the methods of Lagrange and Abel for effecting simUar objects are endoscopic. 
So again, the memoir of Jacobi, ” De Eliminatione,” hereinafter referred to, takes the exoscopic, and the 
valuable “Nota ad Eliminationem pertinens ” of Professor Richelot in Crelle’s Journal, the endoscopic view 
of the subject. In the present memoir (in which the two trains of thought arising out of these distinct 
views are brought into mutual relation) the subject is treated (chiefly but not exclusively) under its endoscopic 
aspect in the second, third and fourth sections, and exoscopically in the first and last sections. 
3 H 2 
