LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 181 



This is another simple form of our fundamental equation. Substitut- 

 ing for m by (67) gives the important equations ^^ 



, 1 6^a p , . 



vV-i0 = -.. (85) 



Let us emphasize once more that all the equations of this section are 

 mere definitions, or purely mathematical deductions, with the sole ex- 

 ception of the one equation which embodies the experimental facts, 

 namely, 



<X>xm = q. 



In conclusion let us consider what is meant by the rotation of the 

 axes in this four-dimensional space. The theory of relativity, as here 

 employed, is equivalent to the statement that our four-dimensional 

 vector equations are invariant in any orthogonal transformation of the 

 axes X, y, z, id. 



The axis let is characterized by the equation -- = e^ = -^ = and 



at ot at 



may be regarded as the four-dimensional locus (" Weltlinie ") of a 

 point at rest. A straight line, making a small angle with this axis in 

 the plane passing through x and ict, is the locus of a point in uniform 

 motion along the x axis. Taking this line as a new axis {ict') and in 

 place of X, a new axis x, perpendicular to y, z, and ict', we have a new 

 coordinate system in which our fundamental equation (69) retains com- 

 plete validity. In other words, as Einstein pointed out, the equations 

 of the electromagnetic field remain true, whatever point is arbitrarily 

 chosen as a point of rest. 



'^* Abraham-Foppl, II, Equations (30 a) and (30 b). 



