MATHEMATICS: F. L. HITCHCOCK 
181 
geometrical applications; Hamilton showed that its properties include 
those of the most general cubic cone.^ This vector cannot be so simply 
expressed in any other algorism. It has, in general, all its axes 
distinct; two of them are on the minimal cone = 0, and are easily 
found. The others are determined by an equation of the fifth degree. 
In developing a classification of various types, I have made com- 
paratively slight use of the technical processes of vector algebra, and 
have based my subclasses on configurations of the axes rather than 
on the possibility of simple algebraic expression. It appears that, 
in the most general type, a normal form of vector is easily obtained in terms 
of the axes themselves. If there are double axes, but no higher axis, 
there is still no particular difficulty, although the normal forms are less 
simple. It is shown that there is only one kind of triple axis; obviously, 
a quadratic vector can have at most two of these. A normal form of 
vector with two triple axes is developed, and has a number of prop- 
erties in the way of symmetry. An axis of the fourth order, on the 
other hand, may be of two kinds. Quadratic vectors with an axis of 
higher order than the third fall naturally into two famihes, accord- 
ing as the axis is of the first, or of the second kind. An axis of the 
first kind is shown to correspond to a double point common to all three 
of the cubic curves defined by equations (1), if we interpret in homo- 
geneous coordinates. The second kind is shown to depend on a par- 
tial differential equation satisfied by the vector Fp. This differential 
condition depends in part on the results of my former papers, where 
the properties of a differential vector have been developed. 
Tests for the existence of axes of any order up to, and including 
the fourth have been given for vectors of any degree whatever. In 
the quadratic case, normal forms are given including all possible types. 
The existence of over one hundred special types makes it very de- 
sirable to have, on the formal side, the means of covering in one compar- 
atively simple algebraic expression as many of these types as possible, — ■ 
and in such a way that their properties are easily correlated. The 
largest number of advantages for this purpose appears to be possessed 
by the form V^p^p, where 0 and 9 denote linear vector functions. In 
Gibbs' notation, this is the same as determining our vector function 
by the cross product of two dyadics. Besides compactness of expres- 
sion, this vector product offers the following advantages: 
1. It is easily interpreted as the most general birational quadratic 
point transformation in a plane. 
2. It differs from a quadratic vector of the most general type only by 
a term in p, which, as already shown, does not alter the axes. 
