48 Prof. F. Slate on a 



time-rate are, when (in^) is constant, 



dQ l _ cu du cffli dv l 



— _m lVl (c 2_ w?)3 /2^ + (cW)!/*;^ ' ^ 4d > 



Defining ??i'= m^v), the last equation is mathematically 

 presentable as a measure of tangential force, 



Yet the first term in the third member is spurious for the 

 physics, so long as {m } ) is constant ; it has its»sole origin in 

 a shift-rate among the frames (U). 



In application to a finite interval, the name rest-system at 

 equation (14) is a collective noun,, comprising selected frames 

 in the possible group (U) ; the earmark of a rest-frame is the 

 transient equality of its permanent (u) and the velocity (v) 

 of (m) at that one epoch. Identify (m^ m ) and («„ v); and 

 let the shift- rate be so regulated that (v = u) perpetually. If 

 the spurious force were real' (i. e., physical), equation (44) 

 would give (T ) correctly, on making the substitutions 



du dv 

 U = V > dt = dt ; mi=m °' ' ' ' < 45 > 



Or it would repeat the essentials of equations (21, 24, 26), 

 on exchanging {vj v c , vj) for (v) ; while due use of the 

 weighting factor harmonizes the forms completely. Thus an 

 assumed constant inertia, plus illusory (imaginary, quaterni- 

 onic) force due to a glossed-over shift, can replace with 

 mathematical precision our assumed variable inertia*, through 

 continuous use of the rest-frames for the several epochs. 



Since the working-speed (yj, v ! ) becomes zero for a rest- 

 frame, that cannot be self-sufficing for the expression of 

 activity. The directest composite plan contains as well 

 frames fixed relatively to (F) f ; but more deviously, rela- 

 tivity reduces the quasi-force (i^dm'/dt) by the factor (c'/rj), 

 coins an unrealized velocity-term (c), attaches it loosely to («,),. 

 and arrives correctly at the activity (A ) of equation (6) essen- 

 tially thus: 



A 9 dm / dm\ / c\ ( dm\ , 



a »=^=t^) = ^H"a)- • • (46) 



* It is apposite to quote Sommerfeld's remark that the element (dr) 

 of proper time " 1st kein vollstandiges Differential." 



t Slate, 'Fundamental Equations,' pp. 147-54. That background of 

 standard frame cannot be quite ignored. Lorentz remarks "Bei einer 

 absoluten Genauigkeit ware auch der Unterscbied der ' Eigenzeit ' der 

 Erde von der Zeit des Sonnensystems zu beriicksichtigen." 



