180 
MATHEMATICS: F. L. HITCHCOCK 
solved completely; these functions constitute a family or set possessing 
some group properties, — just how far they form a group has not been in- 
vestigated, so far as I am aware. 
The axes of the vector function correspond to the singular points 
of (3), if we interpret in homogeneous coordinates. It is well known 
that these are n -\- 1 in number,^ when n is the degree of the 
forms X,F,Z. If = 2 we thus have seven axes, in general. 
The necessity for careful examination and classification of types of 
quadratic vector functions appears from the fact that many differ- 
ential equations like (4) do not yield vectors of the most general kind, 
having all seven axes distinct, but possess multiple or coincident axes of 
all orders up to seven. To take a simple example, ii X = xy, Y = yz 
and Z = zx, the vectors i, j, and k are all double axes, and i + j + k a 
single axis, that is, in homogeneous coordinates, the points (1,0,0,), 
(0,1,0,), and (0,0,1), are higher singularities of (3), and (1,1,1,) is an 
ordinary singularity. By a proper choice of /, that is of 5, we can add 
a term pSdp which shall make Z = 0, and have the equation (4) as 
dy _y {1 — x) ^ 
dx X (y — x)' 
the value of f being — x, and that of d being i. The variables Xfy,z them- 
selves correspond to particular solutions of (3). Four particular solu- 
tions are needed, however, to complete the solution by quadratures; 
hence the rest of the functions of the family cannot be found by mere 
quadratures. 
Again, a quadratic vector may have more than seven axes, but if 
so it has an infinite number, and equation (3) may be divided through 
by a scalar variable. Take for example one of the simplest types 
furnished by the technique of vector algebra, viz., VpVap, or in Gibbs' 
notation p X (a X p) . This vector may be expanded as 
pSap — ap^, 
which differs from the vector a(x^ + y^ + z^) only by the term in p, 
having no effect on the axes. Hence any element of the minimal cone 
p2 = 0 is an axis, and a is the only other axis. 
The most general quadratic vector function may be very elegantly 
defined by a sum of triads, that is, a triadic. A single triad al3y mul- 
tipHed (dot product) by and into p is the same as the Hamiltonian 
vector ^SapSyp. Evidently /5 is an axis. Also, any vector at right 
angles either to a or to 7 is an axis. As a less special example, the 
vector Yqpspt, where qjS,t are constant quaternions, has important 
