MR. SYLVESTER ON A THEORY OF SYZYGETIC RELATIONS. 
54 
Glossary of new or unusual Terms, or of Terms used in a new or unusual sense in the 
'preceding Memoir. 
Allotrioiis. — The allotrious factor to a residue or quotient in the process of common measure ap- 
plied to two algebraical functions is the constant factor of which such residue or quotient must be 
divested in order to become an integral and irreducible function. 
Apocopated. — Applied to a type in the Theory of Cumulants, denotes a type the final or initial 
element of which has been taken away. If both are taken away, the type is said to be doubly 
apocopated. 
Bezoutic. — For definition of Primary and Secondary Bezoutics see first Section. Bezoutiant to 
two functions, each of degree n, is a homogeneous quadratic invariantive function of n variables, 
the form of which serves to assign the index of the scale of the effective intercalations of the real 
roots of the two given functions. 
Bezoutoid. — The Bezoutiant to two homogeneous functions obtained by differentiation from one 
homogeneous function of two variables. The Bezoutoid to a given function of m dimensions in the 
variables is accordingly a quadratic function of (m — 1) variables, the form of which is sufficient for 
determining the number of real roots in the given function. 
Characteristic. — The employment of this w'ord has been avoided in the preceding memoir ; but as it 
contains an idea of capital importance in analysis, and especially in all inquiries of the kind here 
treated of, I subjoin the definition of its meaning. The characteristic of a simple condition of any 
kind is the rational integral function (in its lowest terms) whose evanescence necessarily and uni- 
versally implies and is implied by the satisfaction of such condition. A simple condition has always 
a single characteristic, abstraction being made of the algebraical sign, w'hich remains indetei'minate. 
In like manner, a multiple condition, or a system of conditions, will have for its characteristic a 
plexus of rational integral functions, whose evanescence necessarily and universally implies and is 
implied by the satisfaction of such multiple condition or system of conditions. The number of 
functions in the characteristic plexus will however in general greatly exceed the index of the 
multiplicity of the conditions, and need not always be a unique system. There are however excep- 
tions to this : thus the duplex condition, that a biquadratic function of x shall contain a cubic factor, 
or that a curve of the third degree shall have a cusp, will each be definitely characterized by a 
plexus of two functions, and no more. 
The spirit of the higher analysis resides, and is to be sought for, in the logic of characteristics. 
Co-bezoutiant . — Any homogeneous quadratic function similar in form and in its property of 
invariance to the Bezoutiant. 
Cogredient and Contragredient . — A system of variables is cogredient to another system when it is 
subject to undergo simultaneously therewith linear substitutions of a like kind, and contragredient 
when it is subject to undergo linear substitutions simultaneously therewith but of a contrary kind. 
Combinant. — A function of the quantities appearing in a given set of functions w'hich remains 
unaltered as well for linear substitutions impressed upon the variables as for linear combinations of 
the functions themselves. 
Concomitant. — Nomen generalissimum for a form invariantively connected with a given form or 
system of forms. 
Conjunctive. — A syzygetic function of a given set of functions. Any function which universally, 
MDCCCLIII. 4 B 
