386 



HITCHCOCK. 



PART II. REDUCIBLE VECTORS. 



7. To obtain typical forms for vectors of the third class, which it 

 will be convenient to consider next, much is gained in simplicity by 

 introducing vector multiplication. If we adopt the Hamiltonian laws 

 for i, j, and k, — 



ij = A-, jlc = i, ki = j; ji = — k, kj = — i, ik = — j, 



2-2 = j2 = ^2 = - 1. 



it is well known that vector multiplication is distributive with respect 

 to addition and is associative. It is obviously not commutative. The 

 product of two vectors is, in general, partly a scalar, and partly a 

 vector. These two parts of the product are denoted respectively, 

 by the selective symbols /S and V. We may verify by direct multi- 

 plication that equations (3) are equivalent to the vector equation 



VpF{p) = 



(36) 



It was shown in Art. 2 that, if the number of axes of r(p) is infinite, 

 the left members of equations (3) have a common factor, that is, VpF{p) 

 consists of a scalar factor multiplied into a vector of lower degree. 



I shall now show that reducible quadratic vectors may be thrown 

 into one of three typical forms, according as they possess 



(a) A proper cone of axes, every element of the cone being an axis 

 of the vector. 



(b) A single plane of axes, every direction in the plane being an 

 axis of the vector. 



(c) Two planes of axes, which, as a special case, may be in coin- 

 cidence. 



In each of these cases, there will, in general, be one or more discrete 

 axes not in the plane or cone of axes. 



8. The above subdivision of reducible vectors follows readily from 

 certain properties of homogeneous vectors. 



Theorem II. If a vector jP„(p), whose components are homoge- 

 neous polynomials in x, y, and z of degree n, satisfies the identity 



SpFnip) ^ 



(37) 



