OR OF TERMS USED IN A NEW OR UNUSUAL SENSE. 
547 
n in number, will consist of terms, some of them at least, non-linear, and fewer than n in number. 
These then are the singular cases (or cases of singularity) in the theory of the development of an 
alo’ebraical fraction under the continued fraction form ; and it will be seen that according to this 
definition the case of the development of any proper algebraical fraction in which the degree of the 
numerator is more than one unit below^ that of the denominator, belongs (strictly speaking) to the 
class of singular cases ; and this view of the case supposed is perfectly correct and conformable to 
the analogies of the subject. 
Substitution (linear, similar or contrary) . — A linear substitution is said to be impressed upon a system 
of variables when each variable is replaced by a linear conjunctive of all the variables. The matrix 
formed by the coelficients of substitution arranged in regular order is called the Matrix of Substitu- 
tion, and is of course a square. When two substitutions (impressed in two systems of variables) 
have the same matrix, they are said to be similar and contrary when their matrices are contrary, 
i. e. mutually inverse to each other. W^hen two systems of variables are supposed to be subject to 
the condition that their substitutions are ahvays similar or always contrary, they are said to be 
related or in simple relation, the relation being of cogredience in the one case and of contragre- 
dience in the other. 
When a linear substitution is impressed upon a system of independent variables, a corresponding 
linear substitution is necessarily impressed at the same time upon every complete system of homo- 
geneous combinations [i. e. products and powers and products of powers) of these variables, the 
matrix to which latter substitution will consist of terms which will be functions (depending upon 
the degree of the homogeneous combinations) of the terms of the matrix to the primitive substitu- 
tion. This matrix may be termed a compound matrix, having the primitive matrix for its base. 
If, now, two systems of independent variables are subjeet to be synchronously impressed wdth 
substitutions, the matrices to which (not being both of them simple matrices) have for their bases 
matrices w'hich are either similar or contrary, these two systems wdll be said to be in compound 
relation of cogredience in the one case, and of contragredience in the other. 
Syrrhizoristic. — A syrrhizoristic series is a series of disconnected functions which serve to deter- 
mine the effective intercalations of the real roots of two functions lying between any assigned limits. 
Syzygetic. — A syzygetic function or conjunctive of a number of given rational integral functions 
is the sum of these affected respectively with arbitrary functional multipliers, which are termed 
the syzygetic multipliers. When a syzygetic function of a given set of functions can be made to 
vanish, they are said to be syzygetically related. 
Transform. — Equivalent to the French noun substantive “ transformeeP 
Tape. — The type of a cumulant is the series of the simple elements (or quotients), arranged in a 
fixed order, of which the cumulant is composed. 
Umbral. — The umbral notation is a notation according to w'hich simple quantities are denoted by 
syllables, instead of by single letters (the composition of these syllables being governed by the mode 
in which the quantities which they express are obtained) j and the single letters of such syllables 
are termed umbral quantities or umbrae. 
Weight. — In this memoir (throughout the earlier sections) the weight of any quantity composed 
of the product of the coefficients of any given function or functions of x is used to denote the 
number of roots of x appertaining to the given funetion or functions which must be employed 
to express such quantity. More generally, when dealing with a system of homogeneous functions. 
