WILSON AND LEWIS. — RELATIVITY. 447 



Two 1 -vectors arc tuklcd in the ordiiuiry way by the iwrallelogram 

 law. The same is true of two 2-vcctors if they intersect in a Hne, that 

 is, if they He in the same 3-space (§ 25). It is, however, clear that in 

 four dimensional space it is possible to have tAVO parallelograms which 

 have a common vertex but which do not lie in any planoid, that is, 

 do not intersect in a line. For two such 2-vectors the construction 

 previously given for the sum is not applicable, and it is indeed impossi- 

 ble to replace the sum of the two 2-vectors by a single plane vector. 

 The sum may, however, be replaced in an infinite variety of ways by 

 the sum of two other 2-vectors. For if A and B are any two 2-vectors, 

 and if a and b be two 1-vectors drawn respectively in the planes of A 

 and B, then the 2-\ector a^b = C may be added or subtracted from 

 A and B so that 



A + B = (A + C) + (B - C) = A' + B'. 



The sum of more than two 2-vectors can, however, always be reduced 

 to a sum of two. For if three planes in four dimensional space pass 

 through a point, at least two must intersect in a line. A sum of 

 2-\ectors, which is not reducible to a single uniplanar or simple 2- 

 vector will be called a biplanar or double 2-vector whenever it is 

 important to emphasize the difference. Since the analytical treatment 

 of these two kinds of 2-vectors is not materially different, they will l)e 

 designated by the same type of letters (clarendon capitals). 



A Acctor of the type axbxc will be called a 3-vector. As two planoids 

 which have a point in common, intersect in a plane, a geometric 

 construction for the sum of two 3-vectors may be given in a manner 

 which is the immediate extension of the rule for 2-A'ectors in three 

 dimensional space. The sum of two 3-vectors is always a simple 

 3-vector. 



In respect to rotation and to the classification of lines, planes, and 

 planoids, our four dimensional geometr}^ will be non-Euclidean in 

 such a manner as to be the natural extension of the non-Euclidean 

 geometries of two and three dimensions which have been already 

 discussed. As in two dimensions there were two fixed lines through a 

 point, and in three dimensions a fixed cone, so in four dimensions 

 there will be a fixed conical spread of three dimensions, or hypercone, 

 which separates lines within the hypercone and called (5)-lines, from 

 lines outside the hypercone, which are called (7)-lines. Besides the 

 singular planes which are tangent to the hypercone, there are two 

 classes of planes, namely, (5)-planes which contain a (5)-line, and (7)- 

 planes which contain no (5)-line. Besides the singular planoids which 



