426 PROCEEDINGS OF THE AMERICAN ACADEMY. 



a material particle, and mo represents the mass or energy of the Hohl- 

 raum as it appears to any observer at rest with respect to it. To such 

 an observer the amount of energy traveling in one direction appears 

 equal to that traveling in the opposite direction, and the resultant 

 momentum is zero. To any observer moving with the velocity' v 

 relative to the particle, the momentum is the difference between the 

 momenta which he observes in the two directions, and the mass of 

 the particle is increased in the ratio 1/Vl — «-. These results are 

 all evident geometrically, and follow analytically from (20). 



The Non-Euclidean Geometry in Three Dimensions. 



Geometry, Outer and Inner Products. 



25. We shall now consider a three-dimensional space in which the 

 meaning of points, lines, planes, parallelism, and parallel-transforma- 

 tion or translation are precisely as in ordinary Euclidean geometry. 

 In such a space, in addition to directed segments of lines or one-di- 

 mensional vectors, we have directed portions of planes or two-dimen- 

 sional vectors. Any two portions of the same or parallel planes 

 having the same area and the same sign will be considered identical 

 two-dimensional vectors, briefly designated as 2-vectors. The ordi- 

 nary one-dimensional vectors may be called 1-vectors for definiteness. 

 It is evident that the outer product a>b of two 1-vectors in space is no 

 longer a pseudo-scalar but a 2-vector lying in the plane determined 

 by the two vectors and having a magnitude ec[ual to the area of their 

 parallelogram. 



The addition of two 2-vectors may be accomplished geometrically 

 in the following way. Take a definite segment of the line of inter- 

 section of the planes of the 2-vectors. In each plane construct on 

 this segment as one side parallelograms equal respectively to the given 

 2-vectors. Complete the parallelepiped of which these two parallelo- 

 grams are adjacent faces. The diagonal parallelogram of the paral- 

 lelepiped, passing through the chosen segment, is the vector sum; 

 the diagonal parallelogram parallel to the chosen segment is the 

 vector difference. 



Let us consider the outer product of a 1-vector and a 2-vector,^^ 

 axA. Let A be represented as a parallelogram, and a as a vector 

 through one vertex; the product ax A is the parallelepiped thus 



22 In general 2-vectors will be designated by Clarendon capitals (except in 

 the case of the unit coordinate 2-vectorsJ . 



