WII.SOX AND LEWIS. — KKLVTIVITY. 427 



determined. This outer product a-A, being three-dimonsionul in :i 

 three-dimensional space, is a pseudo-scalar; and diU'orent pseudo- 

 scalars are distinguished only by magnitude and sign. 



If in a^A we regard A as itself an outer product be, the parallel- 

 epiped is written as a (b>c). This same parallelepiped can be re- 

 garded, with tiie possible exception of sign, as (a^b)>c. We shall in 

 fact consider the sign as the same, and write 



a>:(bxc) = (axb)^c = a-bxc, 



so that the associative law holds for the three factors a, b, c. As 

 b\c = — c^b, we shall write ax(bxc) = — ax(cxb), in order that 

 we may keep the law of association for the scalar factor. By succes- 

 sive steps we ma}' write 



axbxc = — b>^axc = be a; 



and hence the outer product of a 1-vector and a 2-vector is not anti- 

 commutative but commutati\e, namely, 



axA = Axa. 



All of these statements are valid in any geometry of the group charac- 

 terized by the parallel transformation. 



26. In the three-dimensional non-Euclidean space, rotation about 

 a fixed point is characterized by the existence of a fixed cone through 

 the point, corresponding to the fixed lines in our plane geometry. 

 An element of this cone always remains an element; points within the 

 cone remain within, and points without remain outside. Besides the 

 lines which are elements of this cone, or parallel to them, there are 

 two classes, namely, 



(5)-lines through the vertex and lying within the cone, and all lines 

 parallel to them, 



(7)-lines through the vertex and lying outside the cone, and all lines 

 parallel to them. 



In like manner planes may be separated into classes. Besides the 

 planes of singular properties which are tangent to the cone along an 

 element, or planes parallel to these, there are 



(5)-planes through the vertex cutting the cone in two elements, and 

 all planes parallel thereto, 



(7)-planes through the vertex and not otherwise cutting the cone, 

 and all parallel planes. The former set, the (5)-planes, contain (5)- 



