WILSON AND LEWIS. — RELATIVITY. 419 



lines is independcMit of any reference to axes, it follows that accel- 

 erated motion must remain accelerated motion rej^ardless of the axes 

 chosen. Moreover, the curvature (i^ 17) of a curve is also independent 

 of any choice of axes. Hence, although it is impossible, as we have 

 seen, to define absolute velocity (that is, all velocity is relative to 

 some assumed set of axes), we may define absolute acceleration if we 

 are willing to define it as the curvature or as any function of the 

 curvature alone. If, however, we wish to use the ordinary measure 

 of acceleration, we must consider the projection of the curvature 

 upon a chosen .r-axis, namely, 



1 dv dv , ,. 



'^=(r:^ydt' °^ dt = ^^-'-^'- 



It is evident that curvature of constant magnitude does not mean 

 uniform acceleration. Indeed if the numerical value of the curvature 

 is constant the point in the .r<-plane must move upon a pseudo-circle. 

 Since the tangent to this curve approaches, but never reaches, the 

 asymptotic fixed direction, it is clear that the velocity of the particle 

 approaches as its limit the velocity of light. For such a motion, the 

 relation between x and / is easilv seen to be 



(1 - vT^ ~ = ^^ or (.T -c,r-{t- cx)2 = R\ 



where R is the radius of curvature, and Ci, C2 are constants of inte- 

 gration depending on the choice of origin for x and t. 



The interval of arc along any (5)-curve is that which was called 

 by Minkowski the Eigenzeit. This quantity is of course invariant 

 in any change of axes. Thus 



Cds = f ^'df — dx"" = f yldr- — dx'\ 



Mechmiics of a Material Particle and of Radiant Energy. 



21. Hitherto we have not assigned to our moving particles any 

 distinguishing characteristics. Let us now consider what follows if 

 we attribute to each particle a mass. It is true, as we shall later see, 

 that the phenomena which must be discussed in connection with the 

 dynamics of a material i)article, even in the case where that particle 

 moves only in a straight line, cannot l)e adequately represented in 

 our two dimensional diagram. Xevertheless those results which can 



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