CONTRIBUTION FROM RESEARCH LABORATORY OF PIH^SICAL 



CHEMISTRY OF THE MASSACHUSETTS INSTITUTE OF 



TECHNOLOGY.— No. 58. 



ON FOUR-DIMENSIONAL VECTOR ANALYSIS, AND ITS 

 APPLICATION IN ELECTRICAL THEORY. 



By Gilbert N. Lewis. 



Received June 28, 1910. 



The great generalization of Einstein, known as the principle of rela- 

 tivity, and its interpretation by Minkowski, have opened a new domain 

 of natural science. The apparent artificiality and paradox of some of 

 the consequences of the relativity theory disappear completely when 

 with Minkowski we regard the science of kinematics as identical with 

 the geometry of four-dimensional space. 



Minkowski ^ and, following him, Abraham ^ have made an important 

 beginning in the use of four-dimensional vector analysis. In general, 

 however, Minkowski used for his more important deductions, not the 

 vectorial method, but the matrix calculus of Cayley. This was un- 

 doubtedly due to the restricted and specialized character of our present 

 vector analysis, for the vector method, permitting as it does a ready 

 survey, and often a visualization of the results to which it leads, has 

 shown its superiority over all other methods in several branches of 

 physics, and there can be no doubt that it is also peculiarly well adapted 

 to the solution of the new problems introduced by Minkowski. 



1 shall attempt to show in this paper what simple changes must be 

 made in our present system of vector analysis to make it immediately 

 adaptable to a space of higher dimensions. Only such changes will be 

 made as are imperatively demanded by the nature of the problem, and 

 these few changes will, I believe, recommend themselves, not only be- 

 cause of the increased generality of the resulting analysis, but because 

 they restore many features of the original, and much neglected, system 

 of Grassmann.*^ 



^ Gottingen, Nachricht., 1908, p. 53. 



2 Rendiconti di Palermo, 30, 1 (1910). 



^ References to Grassmami will be to the edition of 1894, Teubner, Leipzig. 



