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PROCEEDINGS OF THE AMERICAN ACADEMY. 



where u is the relative velocity determined by yp. Or, 



A'B' = AB sech xP = AB Vl — u\ 



That is to say, the distance A'B' between the particles when con- 

 sidered in motion with the velocity u is to the distance AB between 

 the particles when considered at rest as Vl — u^:l. This statement 



embodies Lorentz's theory of the shortening 

 of distances in the direction of motion. 



Consider now (Figure 16) two intersecting 

 (5)-lines along which equal (unit) intervals OT 

 and OT' are marked. If OT is taken as the 

 time-axis, the point M, obtained by dropping 

 from T' the perpendicular T'M to OT, is 

 simultaneous with T'. But the interval OM 

 is greater than OT in the ratio 1 



Figure 16. 



Vl — w 



where u = tanh \}/ is the relative velocity 

 determined by the two lines. Hence a unit time OT' as measured 

 along OT' appears greater with reference to OT than the unit OT 

 itself. This is another statement of Einstein's theorem that unit time, 

 measured in a moving system, is longer than unit time measured in 

 a stationary system. 



All of these special thorems follow directly from the general trans- 

 formation equations (7). We have 



Xi = xi cosh \p — Xi sinh \p, 

 Xi = — .ri sinh \p -\- Xi cosh \p. 



Now substituting 

 u = tanh )/', 



sinh xj/ = u/ Vl 

 1 



w 



cosh i/' = 1 / Vl — u^, 



•Ti 



Xi 



vr=: 

 1 



vr 



u^ 



(Xi — ^^4). 



(a-4 — w.ri) . 



W 



Or, replacing .1-4 by t and Xi by x, we have the fundamental transfor- 

 mation equations of Einstein for the change from stationary to 

 moving coordinates. 



20. Let us next consider instead of a (5)-line any (5)-curve. This 

 will represent the space-time locus of a particle undergoing accelerated 

 rectilinear motion. As the distinction between curved and straight 



