46 Prof. F. Slate on a 



Observations in (F) would show the velocity of (in) relative 



to (0") 



v >i= v '-(i C ii- v >)= v '-v . . . . (37) 



Equation (19) applied correctly to each part of the third 

 member gives for velocities in ({]') auxiliary to (V, v ), 



v c = -; u c = nj. • • ■ (ob) 



ll v uu' 



Then without complications the identity is verified : 



7 K)r' = 7 (W)(l-^)(Y("')TY). • (39) 



Iii verbal form : Direct transition (F, U ; ') and two successive 

 transitions (F, 11'), (U', U' ; ), coincide upon the force finally 

 apparent in (U"). But in terms made consistent with 

 equation (18) as regards all entries under (v"); and only 

 when each apparent force is properly tc weighted." The 

 previous developments furnish plain reasons for both con- 

 ditions. 



Begin with (IT) as observation-frame, and under transferred 

 guidance of equations (13, 2G) construct the provable identity: 



y( «j(i-^)T.' s [ 7 (««)(i-^)]r(»')(i+ "->>:. 



. . . (40) 



Again results agree exactly for direct transition (U', U") 

 and for the two-step process, (U' F) followed by (F, U"). 

 Jn this use, corresponding to equation (38) we must have in 

 their relation to (v) and (U') : 



v f + u' v + u ., 



v c = ,— ; v c == —. . . . (41,i 



Without repetitive detail, therefore, the conclusion can be 

 put generally : When these discriminations about terms are 

 upheld, the calculated net distortion (into apparent force) 

 depends upon the terminal frames alone. Hence it vanishes 

 if the series of transitions closes at the initial frame ; the 

 remark under equation (29) becomes a wider truth*. As 

 bearing upon our immediate purpose, it places all transitions 

 within the group (U) in a comprehensive setting of partial 

 reduction to the same standard (F). 



* A consequence that one might borrow from the familiar theorem on 

 superposition of colinear Lorentz transformations, by assigning present 

 meanings to its symbols. 



