178 PROCEEDINGS OF THE AMERICAN ACADEMY. 



We shall call m the fundamental electromagnetic 1-vector, and show 

 that the four important field equations of Maxwell and Lorentz, as 

 well as other well-known equations, are all contained in the strikingly 

 simple formula, 



OOxrn = q. (69) 



In addition to this equation which states the experimental facts, we 

 have from (54) the important identity 



OxOxm = 0. (70) 



The quantity Oxm^ which might be called the four-dimensional curl 

 of m, is the fundamental electromagnetic 2-vector.2i We will give it 

 the symbol M, 



Oxm = M. (71) 



Expanding Oxm by equation (49) gives 



+ 



The first three terms are evidently equal to the curl of a ; the last 

 three may be put in the form 



I ik^ -^ + 1% -^ + yks -^ - — xk4 ; 

 \ dxi d.V2 d.i'3 d.i\J 



and further collecting terms, and writing ict for .ri, gives 

 Hence, 



M 



. f 1 c)a\ , , , 



V xa + ^ f V (/> + - ^y j xki, (73) 



and thus by equations (61), (65) and (66) 



M = H + E. (74) 



This equation gives a better idea of the physical significance of the 

 2- vector O X m, or M. H is a 2- vector lying wholly in the a\ //, z 

 8-space. E is a 2-vector })erpendicular to the .?', ?/, z 8-space in a plane 

 determined by the 1 -vectors e and k4. We may therefore write, 



21 This is the equivalent of the/ or F of Minkowski. 



