Energy and Relativity. 97 



The first member is on 'the track of the root-idea in 

 Einstein's substitutions ; while the equality leads to the 

 connexions of sums and products, or of quantities and their 

 logarithms, which control the mathematics at many points. 

 Therefore it is not astonishing to find relativity at one with 

 the present argument in building the force (P) and the 

 equation of motion (1) into the foundations. Einstein's 

 velocities actually secure for (P) invariance in form and 

 magnitude throughout the group of frames (U), so long as 

 their derivatives based on a common time-variable are 

 expressed*. Only at the transition to "local time" are 

 the reduction factors combined with this invariant nucleus, 

 which are disentangled and discussed in the previous papers. 

 The task for physics is to trace the simpler elements here 

 plainly indicated through the shifting patterns of a rather 

 kaleidoscopic algebra. 



Retaining at first the general symbol (vj) for a terminal 

 velocity, let us notice the scope of equation (1), some 

 peculiar consequences of it, and its limitations. It implies 

 a total (gross) flux of energy due to the propelling force 

 (P), this being partly diverted from the kinetic energy 

 of (nij) by passive resistances summed in (P). Complete 

 diversion (and conversion) can be entertained as a limit. 

 Secondly, remark how the mass (mi), "weight-mass" and 

 constant in (a), is replaced as a factor of the same accelera- 

 tion by a variable effective inertia in (j>). But the physics 

 of the original form is not modified by this merely 

 mathematical step. In the third place, since the trans- 

 formation involves a terminal velocity essentially, it is 

 intrinsically invalid outside the range (0, Vi) thus marked 

 off, whatever magnitude (y{j may have. Imaginary com- 

 binations which occur beyond those logical limits argue 

 nothing against greater speeds attainable under other 

 physical conditions. Specifically, this holds for light- speed 

 (c) if that happens to enter as a terminal velocity. 



For an interval taken conveniently between rest and any 

 velocity (r), the equation of work in the standard frame (F) 

 leads through equation (!(/>)) to the value 



J W-v 2 dtJ 2 M, r - t ,-y 



w 



Jo \tr-v 



P \ wm 



nuv 



log —7-1= V-Hd — XT I • C 3 ) 



2 & dv 



l dt 



* See (II.), note to eq. (25). 

 Phil. Mag. S. 6. Vol. 41. No. 241. Jan. 1921. II 



