﻿Einstein and Grossmaiui's Theory of Gravitation. 89 



defined as a set of quantities h &v which transform themselves 

 like 



l=3 ...4 

 1=1 ...4 



We can convince ourselves of the fact that L ff „ is trans- 

 formed as dx(r dx v either by direct calculation or by remarking 

 that the stresses, &c. in the electromagnetic field have the 

 same dimensions as stresses, &c. in matter, f. e. in a (viscous) 

 gas. When the distribution law of velocities (f, 7), f) is 

 /(£> Vi ?) &i drj dt,, so that in a space dS there are fd% dr] d£d& 

 molecules with the given velocity, then for X z , the amount 

 of momentum carried across unit of an area perpendicular 

 to the axis of X, we find 



Similarly, for X y 



m dx dx 



>2~r*'dt'dt 



X„ = J/Vf^< 



11 dx dy 

 ^~^-v*'di'dt' 



and so on. 



We see the products dx dx, dxdy, that is dx G dx v , come in, 

 and entering into details we could prove that indeed the 

 components of a tensor of stresses, momenta, and energy 

 transform themselves as dx a dx v . 



This property is of great use when we wish to write 

 equations in a form that is invariant against transformations. 

 dt causes the four quantities 



•££? (-=1,2,3,4) 



v O x v 



to transform themselves as dxa, that is, they are the com- 

 ponents of a four-dimensional vector (which might be called 

 a tensor of the first order). Therefore, when they are equated 

 to the components of: a vector, such as the force (F ff ), then 

 both members of the equations 



"dxp 



= Fa 



are transformed in the same way, and the equation persists 

 in the same form : 



