392 PROCEEDINGS OF THE AMERICAN ACADEMY. 



but also in Euclidean space. We are led further to a consideration 

 of certain vectors of singular character. The study of the singular 

 plane leads to the brief consideration of another interesting and im- 

 portant non-Euclidean plane geometry. 



Passing to the general case of four dimensions we shall meet further 

 new types of vectors, and shall attempt even here to facilitate as far 

 as is possible the visualization of the geometrical results. We shall 

 continue further the analytical development, and in particular con- 

 sider the properties of the differential operator quad. In this con- 

 nection a very general and important equation for the transformation 

 of integrals is obtained. The idea of the geometric vector field will 

 then be introduced, and the properties of these fields will be taken up 

 in detail. 



The subject of electromagnetics and mechanics is prefaced with a 

 short discussion of the possibility of replacing conceptually continuous 

 and discontinuous distributions by one another, and we shall point 

 out that in one important case such a transformation is impossible. 

 The science of electromagnetics is treated both from the point of view 

 of the point charge and from that of the continuous distribution. 

 In both cases it is shown that the field of potential and the field of 

 force are merely the geometrical fields previously mentioned, except 

 for a constant multiplier. Particular attention is given to the field 

 of an accelerated electron,^ and in this field we find that the vectors 

 of singular properties play an important role. With the aid of these 

 vectors the problem of electromagnetic energy is discussed. The 

 science of mechanics, which is treated in a fragmentary way in some 

 preceding sections, is now given a more general treatment, and the 

 conservation laws of momentum, mass and energy are shown to be 

 special deductions from a single general law stating the constancy of a 

 certain four dimensional vector, which we have called the vector of 

 extended momentum. Finally it is pointed out that this last vector 

 gives rise to geometric vector fields which can be identified with the 



3 There seems to be a widespread iinpressiou that the principle of relativity 

 is inadequate to deal with problems involving acceleration. But the essential 

 idea of relativity can be expressed by the statement that there are certain 

 vectors in the geometry of four dimensions which are independent of any 

 arbitrary choice of the axes of space and time. Those problems which involve 

 acceleration will be shown to possess no greater inherent difficulties than 

 those that involve only uniform motion. It is, moreover, especially to be 

 emphasized that the methods which are to be employed in this paper necessi- 

 tate none of the approximations that are commonly employed in electro- 

 magnetic theorj'. Such terms as "quasi-stationary," for example, will not 

 be used. 



