544 GLOSSARY OF NEW OR UNUSUAL TERMS, 
and suHect to no cases of es^cepUon, vanishes when a eertain number of other functions aU vanish 
together must be aconjunctive (he. a syaygetic function), or a root of a conjuncUve of such tocmns. 
But if its vanishing is subject to oases of exception, then aU that can be predicated of it is that it is 
syzygetieally related to such functions, but it may, and usually does happen, that it will be syzy- 
getically related to them in more than one way. 
Contravariant.-fe function which stands in the same relation to the primitive ftuction from which 
it is derived as any of its linear transforms to an inversely derived transform of its pnmime. 
Caaariant.-A function which stands in the same relation to the primitive function from which it 
is derived as any of its linear transforms to a similarly derived transform of its primitive. 
Cumulant.-Tbe denominator of the simple algebraical fraction which expresses the value of an 
improper continued fraction. See TypCj infra. 
Determinant.— word is used throughout in the single sense, after which it denotes the alter- 
nate or hemihedral function the vanishing of which is the condition of the possibility of the coexist- 
ence of a system of a certain number of homogeneous linear equations of as many variables. 
Dialytic.—li there be a system of functions containing in each term different combinations of the 
powers of the variables in number equal to the number of the functions, a resultant may be formed 
from these functions by, as it were, dissolving the relations which connect together the differen 
combinations of the powers of the variables, and treating them as simple independent quantities 
linearly involved in the functions. The resultant so formed is called the Dialytic Resultant of the 
functions supposed; and any method by which the elimination between two or more equations can 
be made to depend on the formation of such a resultant is called a dialytic method of ehmina ion. 
In such method accordingly the process of elimination between equations of a higher degree thaii 
the first is always reduced to a question of elimination between equations which are of the hrst 
degree only. 
DiscTiminant.-The resultant of the n differential coefficients of a homogeneous function of « valu- 
ables. See Resultant} infra. 
m,mnctive.-k disjunctive equation is a relation between two sets of quantities such that each 
one of either set is equal according to some unspecifled order of conne.xion with some one of e 
other set. 
Effectire scale of intercalations is the series of the real roots of two functions of a- written in 
order of magnitude after repeated processes of removing pairs of roots belonging to either tlie 
same function (when not separated by roots of the other function) i the roots of the two functions 
follow each other alternately. 
every homogeneous function of any number of variables i of the degree 
where mnJ are any two integers, may be formed (as shown in the Calculus of Forms, Section T.) a 
covariantive function of the degree m and of y. variables [where p is the number of permutations 
that can be obtained by dividing m! into i parts (zeros admissible)], in which all the coefficients are 
numerical multiples of the given coefficients; covariants so formed maybe termed effluents of their 
primitive. An example of this occurs in the foot note to Section V. p. 522, where the quantity there 
called Q is a quadratic effluent of the J acobian. 
Element.— h. simple component of the type to a cumulant. See Cumulant, sujira. 
