dt = 



330 Sir Joseph Larmor 



The complete circumstances of the orbits in a field of force of 

 potential energy —V per unit mass (in a gravitational field V is 

 TiSm/r) are condensed into the single variational Least Action 

 equation of Lagrange-Hamilton, 



with integration between limits of time fixed and unvaried. This 

 suggests comparison with the equation for the shortest or most 

 direct path in a modified fourfold involving Euclidean space com- 

 bined with a measure of time varying from place to place: for 

 that equation is 



Sjda=0 where Sa^ ^ Sx'- + Sy^ + Sz^ - c'^Bt^ 



in which c' is a function of x, y, z. Let us write 



C'2 = C2 (1 -f K), 



where K _is very small on account of the greatness of c. The 

 equation is now 



or approximately up to the fourth order 



■&--i---im-m-m. 



dt^ 0. 



The time-limits being unvaried the first term — c^ can be omitted : 

 thus this variational equation of most direct path coincides with 

 the previous orbital equation if 



- |Zc2 = F. 



Thus the forces are absorbed into a varying scale of time; and the 

 motion being now free under no force, the orbit is, as was antici- 

 pated, a geodesic or straightest path. The orbits have become 

 however straightest paths, not in their original Newtonian separ- 

 ated space and time, but in the uniform space-time fourfold of 

 relativity as slightly deranged by the not quite constant scale of 

 time. 



Thus the orbits in any field of attraction have actually been 

 fitted into the mixed space- time frame of electrodynamic relativity, 

 at the expense of doing slight violence to that frame, by making 

 the measure of time vary from place to place while the positional 

 specification remains uniform. 



But this transformation does more than is needed. It ought 

 somehow to be restricted to the one universal force of nature, that 

 of gravitation with its inverse-square law. It is here that the 

 special feature of the Einstein theory seems to come in. For 



