New Heading of Relativity . 37 



its obligatory type. The artificial infinity of (m 2 ) for (v=c) 

 is requisite to preserve the zero-acceleration at (P = R) with 

 finite (m^. Within the freedom to utilize epoch-values for 

 (nil), choose a series that gives perpetual equality of (?>i, nil). 

 That device amounts to completing equation (10), and 

 establishing an exact step-by-step agreement in force-magni- 

 tude with equation (5). Yet the two plans do not quite 

 coalesce ; (dm) can never be primarily an increment of any 

 constant (mi), also residual divergence from equations (3, 6) 

 must continue. The Principle of vis viva is differentially 

 valid ; each constant (m^ absorbs a work-element wholly 

 into change of kinetic energy. Remark finally that assign- 

 ment of permanent value (m ) to (mj) brings in a significant 

 instantaneous relation 



T„ = 7 (r)P [m^mo] (11) 



This line of possible connexion has been dwelt upon as 

 preliminary at more length than its simple thought would 

 justify, did it not strike centrally at the reasons — often left 

 elusively subtle — which necessitate certain transitions peculiar 

 to relativity. Its plain bearing will not be overlooked, it 

 can be hoped, upon the duplicated forces of "Minkowski 

 type " and of "Newton type" ; nor upon the introduction 

 of "proper time" as well as "local time." Both of which 

 discriminations are given effect differentially, on the model 

 of equation (11) *. 



Associated with the standard frame (F) are a group (U) 

 of comparison frames ("legitimate frames") having un- 

 accelerated translations (u) relative to it, {u,v ) being colinear, 

 and for standard sign codirectional. Transfers between (F) 

 and (U) disturb fundamental expressions ; but we shall test 

 there what degree of approach can be disclosed nevertheless 

 to that permanence of type which is relativity's victory. 

 Velocity and tangential acceleration belonging to (F) will 

 show in any (U) the magnitudes 



d'= v —u ; -— = -5T- [00 is vector-constant.] 



(XL CLL 



* Relativity has invented manipulation of factors like (y{v)), where it 

 needs to add or to suppress them, with admirable success. But no more 

 will be expected here than to indicate clues at important turns of the 

 discussion, after which calculative detail is read with new meanings in 

 mere routine. Moreover, the points made can appeal to those only 

 who judge the argument with an interest due to knowledge of the 

 subject. 



