548 
GLOSSARY OF NEW OR UNUSUAL TERMS. 
the weight of a quantity may be defined with respect to any selected variable therein as the sum 
of the weights in respect to such variable of the several coefficients of which the quantity is com- 
posed (the weight of each several coefficient meaning the index of the power of the selected variable 
in that term of the given function or functions which is affected with such coeflBcient). These two 
definitions of weight may be perfectly well reconciled with each other by understanding the weight 
of a quantity formed from the coefficients of a function or system of functions of x to mean the weight, 
in respect to unity, of such quantity when the given functions are treated as homogeneous functions 
of X and 1. 
Zeta . — The symbol f (preceding a row of bracketed terms) is used to denote the product of the 
squared differences of the terms which it affects. 
[ ] . A bracket of this form, when inclosing a superior and an inferior row of terms m and n in 
number respectively, indicates the mn products of the differences obtained by subtracting each term 
in the second row from each term in the first row; when enclosing an arrangement of terms in a single 
line, it is used to denote the cumulant of which such an arrangement is the type. 
CONTENTS. 
Page 
Introduction 407 
Section I. — On the complete and simplified residues generated in the process of developing under the 
form of a continued fraction, an ordinary rational algebraical fraction 415 
Section II. — On the general solution in terms of the roots of any two given algebraical functions of x 
of the syzygetic equation, which connects them with a third function, whose degree in (x) is given, 
but whose form is to be determined 4S3 
Section III. — On the application of the Theorems in the preceding Section to the expression in terms 
of the roots of any primitive function of Sturm’s auxiliary functions, and the other functions which 
connect these with the primitive function and its first differential derivative 456 
Supplement to Section III.— -On the Quotients resulting from the process of continued division ordi- 
narily applied to two Algebraical Functions in order to determine their greatest Common Measure... 467 
Section IV. — On some further Formulae connected with M. Sturm’s theorem, and on the Theory of 
Intercalations whereof that theorem may be treated as a corollary 4S0 
Supplement to Section IV. — Development of the method of assigning a superior and inferior limit 
to the roots of any algebraical equation 
496 
Section V.— On the Theory of Intercalations as applicable to two functions of the same degree, and on 
the formal properties of the Bezoutiant with reference to the method of Invariants 510 
General and Concluding Supplement. — General Theorem connecting the Partial Quotients with the Con- 
vergents to an Unlimited Arbitrary Rational Integral Algebraical Continued Fraction, and on the 
Conversion of a Series of Partial Fractions into a Continued Fraction 535 
Errata -542 
Glossary of new or unusual Terms, or of Terms used in a new or unusual sense in the preceding Memoir 543 
