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XVIII. On a Theory of the Syzygetic* relations of two rational integral functions, 
comprising an application to the Theory of Sturm’s Functions, and that of the 
greatest Algebraical Common Measure. By J. J. Sylvester, M.A. Dub., F.R.S., 
Barrister at Law, and formerly Professor of Natural Philosophy in University 
College, London. 
Received and Read June 16 , 1853 . 
Introduction. 
' How charming is divine philosophy ! 
Not harsh and crabbed as dull fools suppose. 
But musical as is Apollo’s lute. 
And a perpetual feast of nectar’d sweets. 
Where no crude surfeit reigns !” — Comus. 
In the first section of the ensuing memoir, which is divided into five sections, I con- 
sider the nature and properties of the residues which result from the ordinary process 
of successive division (such as is employed for the purpose of finding the greatest 
common measure) applied tof{x) and (p{x), two perfectly independent rational integral 
functions of x. Every such residue, as will be evident from considering the mode in 
which it arises, is a syzygetic function of the two given functions ; that is to say, each 
of the given functions being multiplied by an appropriate other function of a given 
degree in x, the sum of the two products will express a corresponding residue. These 
multipliers, in fact, are the numerators and denominators to the successive convergents 
i. 
® fx ®^pf6ssed under the form of a continued fraction. If now we proceed a priori by 
means of the given conditions as to the degree in {x) of the multipliers and of any 
residue, to determine such residue, we find, as shown in art. (2.), that there are as 
many homogeneous equations to be solved as there are constants to be determined; 
accordingly, with the exception of one arbitrary factor which enters into the solution, 
the problem is definite ; and if it be further agreed that the quantities entering into 
the solution shall be of the lowest possible dimensions in respect of the coefficients of 
/and (p, and also of the lowest numerical denomination, then the problem (save as to 
the algebraical sign of plus or minus) becomes absolutely determinate, and we can 
assign the numbers of the dimensions for the respective residues and syzygetic mul- 
tipliers. The residues given by the method of successive division are easily seen not 
Conjugate would imply something very different from Syzygetic, viz. a theory of the Invariantive properties 
of a system of two algebraical functions. 
MDCCCLIII. 
3 H 
