EINSTEIN'S LAW OF GRAVITATION 241 



where ds is the interval and x^ x 2 , x 3 , x^ are coordinates. 

 If we make a mathematical transformation, i.e., use an- 

 other set of axes, this interval would obviously take the 

 form 



+ ff22<t* 2 2 + 



2g l2 dx 1 dx 2 + etc., 



where x^ x 2 , x z and Jtr 4 are now coordinates referring to 

 the new axes. This relation involves ten coefficients, the 

 coefficients defining the transformation. 



But of course a certain dynamical value is also attached 

 to the g's, because by the transfer of our axes from the 

 Galilean type we have made a change which is equivalent 

 to the introduction of a gravitational field ; and the g's 

 must specify the field. That is, these g's are the expres- 

 sions of our experiences, and hence their values cannot 

 depend upon the use of any special axes ; the values must 

 be the same for all selections. In other words, the expres- 

 sion of the facts of gravitation is really a statement in- 

 volving a relation between the g's ; and this expression 

 must be the same for all sets of coordinates. There are 

 ten g's defined by differential equations ; so we have ten 

 covariant equations. Einstein showed how these g's could 

 be regarded as generalized potentials of the field. Our 

 own experiments and observations upon gravitation have 

 given us a certain knowledge concerning its potential ; 

 that is, we know a value for it which must be so near the 

 truth that we can properly call it at least a first approxi- 

 mation. Or, stated differently, if Einstein succeeds in 

 deducing the rigid value for the gravitational potential in 

 any field, it must degenerate to the Newtonian value for 

 the great majority of cases with which we have actual 

 experience. Einstein's method, then, was to investigate 

 the functions (or equations) which would satisfy the 

 mathematical conditions just described. A transforma- 

 tion from the axes used by the observer in the falling box 



