446 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then 



mv = man ( — ki sin 7it + k2 cos nt) + mki, 



f = , = — man- Cki cos 7it + ko sin ?«i) = — mn-Vs, 

 at 



where r^ is the component of r on the two-dimensional "space." 

 The force is directed toward the center, as usual. It may be observed 

 that if in general the force is central, the moment of momentum is 

 constant. For if 



— (mv) = f , Tsx— (mv) = -7, (rsxmv) = r^xf = 0. 

 at at at 



That the rate of change of moment of momentum is equal to the mo- 

 ment of the force is therefore a principle which holds in non-Newtonian 

 as in ordinary mechanics. 



The Non-Euclidean Geometry in Four Dimensions. 



Geometry and Vector Algebra. 



37. Consider now a space of four dimensions in which the elements 

 are points, lines, planes, flat 3-spaces or planoids, and which is sub- 

 ject to the same rules of translation or parallel-transformation as two 

 or three dimensional space. If a and b are two 1-vectors, the product 

 axb is a 2-vector, that is, the parallelogram determined by the 

 vectors. The parallelograms axb and bxa will be taken as of 

 opposite sign, but otherwise equal. The equation axb = ex- 

 presses the condition that a and b are parallel. If C is any third 1- 

 vector, not lying in the plane of a and b, the product axbxc, 

 which is now itself a vector will represent the parallelepiped deter- 

 mined by the three vectors. The condition axbxc = there- 

 fore states that the three 1-vectors lie in a plane. If now d is a fourth 

 1-vector, not lying in the 3-space or planoid determined by a, b, C, 

 the product axbxcxd will represent the four dimensional parallel 

 figure determined by the vectors. This product is a pseudo-scalar 

 of which the magnitude is the four dimensional content of the 

 parallel figure. The condition axbxcxd = shows that the four 

 vectors lie in some planoid. In all these outer products the sign is 

 changed by the interchange of two adjacent factors, as in the case of 

 lower dimensions. Moreover, the associative law, the distributive 

 law, and the law of association for scalar factors will also hold, as is 

 evident from their geometrical interpretation. 



