166 PROCEEDINGS OF THE AMERICAN ACADEMY. 



In the second section several of the most useful formulae of four- 

 dimensional vector analysis will be presented, and the last section will 

 be devoted to some applications of these formulae in electromagnetic 

 theory and the theory of relativity. 



The Vector Analysis of Three Dimensions, 



The simplest type of quantity which is distinguished from others of 

 its class by magnitude and direction is the familiar line- vector, a one- 

 dimensional quantity which we shall call a vector of the first order, or 

 in brief, a 1 -vector. 



Just as two parallel line- vectors of the same length are regarded as 

 equal, so two parallel plane surfaces of the same area are also consid- 

 ered equal. A plane area ^ constitutes a vector of the second order, 

 or a 2 -vector. 



In general in a space of n dimensions we may distinguish 0-vectors 

 or scalars; 1-vectors, 2-vectors, 3-vectors, etc., up to the ??-vectors, 

 which, like the 0-vectors, have no direction and may therefore be called 

 pseudo-scalars. 



In three-dimensional space the only true vectors which exist are 

 1-vectors and 2-vectors. Moreover, every 2-vector determines uniquely 

 the 1 -vector normal to it. In common vector anal)^sis the 2-vector is 

 regarded as equivalent to and replaceable by its normal 1 -vector of the 

 same magnitude, and therefore this analysis deals solely with 1-vectors. 

 This simplification has certain obvious advantages which, however, 

 are for the most part superficial. In some cases moreover it leads to 

 difficulties.^ In any case it must be abandoned when we pass to space 

 of higher dimensions, where a 2-vector no longer uniquely determines 

 a 1 -vector. 



Our first departure, then, from common vector analysis will consist 

 in distinguishing between vectors of different orders. A 1 -vector will 



* For simplicity we may deal only with Ihe straight vectors (straight line, 

 plane surface, etc.) since any curve tcrniinating in two points may be regarded 

 as eiiuivalent to the straight line terminating in the same points, and a curved 

 surface terminating in a 'plane closed curve, as equivalent to the plane area 

 having the same boundary. Such a vector as a curved surface bounded by a 

 closed curve which does not lie in a plane we shall not consider here. 



^ See, for example, the discussion of scalars and pseudo-scalars in Abraham- 

 Foppl, Theoric dcr Elektrizitiit, p. 22-23. We shall have frequent occasion 

 to cite this standard work, which contains an admirable presentation of the 

 current system of vector analysis, as well as of electrical theory. References 

 arc to the edition of 1904 (Teubner, Leipzig). 



