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MATHEMATICS: F. L. HITCHCOCK 
3. The properties of a quadratic vector are made to depend on those 
of Knear vectors. 
4. For the study of differential equations it is especially suited since 
it is a vector product. Equation (3) takes the factored form 
S(f>p9pYpdp = 0, or in Gibbs' notation (<j>p X dp) . (p X dp) = 0. 
5. Three of its axes are zeros, that is, for three values of p the vector 
vanishes in all its components. 
It appears, therefore, as a problem of importance to determine how 
far the various possible configurations of axes, in special types of quad- 
ratic vectors, are included among possible configurations of the axes of 
V(f)pdp. This is the same as the problem of determining with what 
exceptions a quadratic vector of whatever type or sub-t)^e can be 
written in the form 
Y<j>pdp + pS5p. 
At this point of the investigation a certain difficulty presented itself. 
It is easy so to chose 8 that the resulting quadratic vector shall have 
three zeros, distinct or multiple ; if, then, the vector does not fall into a 
uniplanar, i.e., a binomial form, it is possible to factor vectorially into 
Yct)pdp. But a binomial quadratic vector cannot be so factored, hence 
the necessity of examining a very large number of choices of d to find 
those which do not yield a binomial. For most types where such a vec- 
tor d can be found, I have contented myself with giving the value of the 
resulting 5, since its accuracy, when found, is easy to verify. In the 
cases where no value of 6 can be found, I have, of course, demonstrated 
the impossibility. 
The final result is, on the whole, highly satisfactory. It appears 
that the form V</)p^p + pS5p includes all types of quadratic vectors except 
two simple sub-types both belonging to the family hazing a higher axis of 
the second kind. Normal forms for these two very exceptional types 
have been given. 
The existence of these exceptions is due to the fact that, as the form 
of the vector grows more and more restricted, the possible choices of b, 
which avoid the binomial, decrease in number. Thus in general there 
are thirty-five possible values of 5; but if the determinant of the com- 
ponents of a set of three axes vanishes, the number falls to thirty-one; 
and the occurrence of multiple axes also reduces the number. The 
wonder appears to be, not that there are exceptions, but that there are 
so few, and these so simple. 
Various properties of quadratic vector functions appear by virtue of 
the normal forms which characterize their types. Some of these bring 
