656 Prof. F. Slate on Force- Transformation, 



The special auxiliaries (vj, v a ), which in our rendering 

 replace the Einstein velocities, can be brought formally 

 closer to the latter under a similarly expanded notation. 

 Instead of accepting them initially as " distorted " (but 

 c.g.s.) velocities *, throw each of their defining ratios into 

 terms o£ its own "distorted" length and time : 



, v — u da? ' Vo +u 



Vc = — - = -77T > Vc = ■ 1 



1 lll'o dt c - UV 



dx { 



di, 



then 



[ (14) 



dx t 



dxj 



^i = y(uKv '+u), -^= 7(w )(, -u). 



In effect, details are here borrowed from relativity 

 again another conception of them is attached t< 

 notation : 



x c ' = y(u){x —ut ) ; t ' == 7 (u) \t — - 2 x J 5 



x e = y(u){x ' + ut ') ; t c == y(ti) (tj + 1 x 'j. ■ 



buf 

 the 



(15) 



t*r== 



Let now the more general (z , ?') be particularized 

 provisionally as (y(v ), 7(^0') )• I* is directly provable that 



dto _ rft,' , = jftJ__ '/A • 



KO~yO</)' "7(^) 7(^)' 



^T = ^7 / [>o=<|i • (16) 



which is our version of '' proper-time invariance " for 

 frames (U). At this point, as at others, a group of four 

 quantities doubles relativity's pair, and a pair (dr, dr) 

 distinguishes where relativity does not. Both duplications 

 are referable to combining the two aspects of " simul- 

 taneous ? ' : it is evident that, whenever observed (v \ v ) 

 coincide with specified series (y c \ Vc), the values calculated 

 for the other frame will coincide also with the (u , vj) cor- 

 responding to such (v e ', v c ). 



Differentiate equations (14), noting equations (16) and 

 previously demonstrated equalities t ; which gives 



7 2 M-^ = 7 3 ("»)^ = *(".)-& -•/(•..) 57 -sr. 



"1 



Y 2 W) 



,, dvj 



dr' 



tXrJl 



dv 

 dto 



dv c 



T' 



= 7W S? =7 ^ ) ^ =m y(uyj 



y (i7) 



* See (L), pp. 436, 438 ; fll.\ pp. 40, 4]. 



f In (JL), eq. (19, 25). Consider eq. (21, 24) there and eq. (17) here 

 for features of likeness and difference. A " reduction-factor " and a 

 " distorted time " are equivalent in operation. 



