17G PROCEEDINGS OF THE AMERICAN ACADEMY. 



Any of these operators may be applied to a scalar or to a vector of any 

 order. 



Two other Important operations are connected with O^ by the formula 

 analogous to (32) : 



O (O xa) = OOxa = 0(0 a) - O'a. (53) 

 And we have here also two important identities 



Ox(Oxa) = 0, (54) 



. Ox{C4>) = 0. (55) 



In the same way that we obtained equations (28), (29), (30), we find 



0(</.a) = «^(Ca) + a (0«A), (56) 



Ox(<^a) = (Cxa) + (Cc^) xa, (57) 



Ox(axb) = bx (Cxa) - ax (Oxb). (58) 



These equations will suffice to illustrate how readily the generalized 

 vector analysis of the preceding section may be applied in a space of 

 any dimensions. 



Some Applications of Four-Dimensional Vector Analysis in the 

 Theory of Electricity. 



The principle of relativity as interpreted by Minkowski can be 

 summed up in the statement ^^ that a Euclidean four-dimensional space 

 is determined by the coordinates, a:, y, z, and ict, where / is the unit of 

 imaginaries, /y/_ i ; and c is the velocity of light. The whole science 

 of kinematics is merely the geometry of this four-dimensional space. 

 As Minkowski himself has shown, there is no domain in which this new 

 conception is more fruitful than in the science of electricity and 

 magnetism. 



Let us consider a system composed of electric charges moving in 



free space. The density of charge at any point we may call - in elec- 

 tromagnetic units, and if we call the velocity of the charge v, then 

 -V represents the current density at a point. This 1 -vector - v lies 

 wholly in the 3-space a; i/, z. 



^8 This statement is subject to certain restrictions that we will not discuss. 



