OR OF TERMS USED IN A NEW OR UNUSUAL SENSE. 
545 
Emanant. — The result of operating any number of times (suppose i times) upon a given homo- 
geneous function of any number of variables x, y, z.,.t with the operative symbol 
( ,d ,d ,d . ,id\ 
Vdso'^ydy'^^j2^"’^^JtP 
is called the ith emanant of the function operated upon. Every emanant is a covariant to its primi- 
tive, the new variables sd, y', being cogradient with the variables x, y, s...t with which they 
are respectively associated. Esi+i/? Egi+e-ip, page 522, are emanants of/and<p. The process of 
emanation is one of incessant occurrence in the theory of invariants. When the order of the emana- 
tion is the same as the degree of the function (supposed to be rational and integral) from which 
the emanation proceeds, the form of the original function is reproduced in the final emanant, the 
names only of the variables being changed. 
Endoscopic, Exoscopic, — When the coefficients of the functions concerned in any investigation are 
regarded as integral indecomposable monads, the method is called exoscopic, and endoscopic when 
the coefficients are treated with reference to their internal constitution as composed of roots or other 
elements. 
In addition to the examples in the foot note to Section 1, these words have a marked and most 
important application in the theory of Invariants, especially of two variables. 
Form. — Any function may be regarded as an opus operatum ; the matter operated upon being the 
variables, and the substance of the operations being the form, which resides in the function as the 
soul in the body. A form is always common to an infinity of functions, but for greater brevity may 
be and frequently is called by the name of some specified function in which it is contained. 
Fundamental. — The fundamental scale of a system of Invariants or Concomitants is a set of the 
same, whereof every other is a Rational Integral Function. 
Hessian or Hessean, named after Dr. Otto Hesse, of Konigsberg (the worthy pupil of his illus- 
trious master, Jacobi, but who, to the scandal of the mathematical world, remains still without a 
Chair in the University which he adorns with his presence and his name), is the Jacobian to the 
differential coefficients of a homogeneous function of any number of variables. It is to a Jacobian 
what a Bezoutoid is to a Bezoutiant, or a Discriminant to a Resultant. 
Hyper determinants. — See Memoir of Mr. Cayley, Cambridge and Dublin Mathematical 
Journal, May 1845, and Crelle’s Journal of about the same date. 
Improper, continued fraction is a continued fraction differing only from an ordinary one in the 
circumstance of negative signs being substituted for positive signs to connect the terms. 
Inertia. — The unchangeable number of integers in the excess of positive over negative signs which 
adheres to a quadratic form expressed as the sum of positive and negative squares, notwithstanding 
any real linear transformations impressed upon such form. 
Intercalations. — The theory of intercalations is the theory of the relative distribution of the real 
roots, or point-roots, of two or more equations, but in this theory the number of roots mutually 
interposed is to be taken only with reference to the number 2 as a modulus. 
Invariance. — The property (under prescribed or implied conditions) of remaining invariable. 
Invariant. — A function of the coefficients of one or more forms which remains unaltered when 
these undergo suitable linear transformations. 
4 B 2 
