Proper Time, and Fresnel's Coefficient. 659 



the answer to tlie question : Which relative speed (v) is 

 phvsically responsible for the resistance (mjcv 2 ) ? 



The manifold bearings of the force (Ti) justify adding a 

 word about its impulse ; and transitions between frames 

 (F, U) where it occurs. As in equation (23), take the 

 interval iu, v ) in (F), important in corresponding to the 

 interval (0, v — u) or (0, r/) in (U). Adjust a velocity (v u ) 

 in (XJ) to meet the condition making momentum-changes 

 permanently equal; as (vj) affected an energy-change : 



miv u =vi 1 (y{r )v — ry(u)u) (26) 



Then the equal time-rates entail 



the tangential accelerations and their forces thus measured 

 in (F, U) are equal ; and the activities as well, 



v u T l = T 1 (y(v.)v.-y(u)u). .... (28) 

 Consequently, 



T^ — l'u) = TYy(» = ry( U y\\[v — (r — U)~\ 



l\v u - Tj7(?0 (r - u) = T 1 r ( 7 (r ) - y(u) ). 



(29) 



Understanding that (??? ) in (TV) is replaced by (wj) for this 

 occasion, the work-relation appears* : 



C u c Vc ' c v ° 



I '\\v u dt -y{u)\ TJv 'dt =\ T x Vo(y(v )-y(uy)dto 



This result does something to enlarge command of inter- 

 dependence between (F, U). But its better service, perhaps, 

 is to enforce again two dynamical ideas that pervade these 

 investigations : first, that the attacks through energy and 

 through momentum, though reconcilable, are not entirely 

 congruent ; and secondly, that the treatment of variable 

 inertia breaks away from what suffices for constant inertia. 

 Each term in the first member builds upon its own equal 

 acceleration (dvjdto, dv \dt ) in (F, U). This is not on the 

 surface true of the second member, since (TV'i") belongs to 

 the third member oE equation (8), and Q£\V„) to its fourth 

 member t- The above expansion repeats for an observatioh- 



* Relying on (I.), pp. 436, 438 ; or on (II.), p. 44. 

 t Cf. the earlier comment ; (II.), pp. 38, 39. 



