﻿504: Prof. E. V. Huntington on a New 



Corollary. If a series of points in S' lie on a circle, 

 according to the observers on S', their images in S will 

 appear to observers in S to lie on an ellipse, the minor axis 

 lying in the direction of motion, and the ratio of the axes 

 being 1/k. 



16. Theorem 3. Let two observers, A and B, on S agree 

 to observe the coordinates and the clock readings at the points 

 of S' which are opposite them " at a specified time." Suppose 

 when A's clock reads t , the point opposite A has an abscissa 

 =Xi and a clock-reading = £/ ; and when B's clock reads 

 t , the point opposite B has an abscissa = £c 2 ' and a clock- 

 reading = t 2 '. Then it will appear that 



. i j. i v / , / f \ 



H ~ l \ — — ~2 \ d '2 ~ X l)' 



v 



This theorem tells us that two clocks on S' which are 

 synchronous when tested by the standards adopted on S', are 

 not synchronous when observed from 8 — the forward clock 

 being slower than the other by an amount proportional to the 

 distance between them. — This result, like the last, is entirely 

 independent of the relative rates of the clocks at and 0'. 



The composition of velocities. 



17. Further, we suppose that an aeroplane passes over platform S with 

 a given observed velocity u, and inquire what the observed velocity of the 

 aeroplane will be with respect to platform S'. For simplicity we confine 

 ourselves to the case of motion in a straight line parallel to the line of 

 motion of the platform S'. 



Theorem 4. If u — the observed velocity of the aeroplane as computed 

 by two observers on S, and u = its observed velocity as computed by two 

 observers on S' , then 



-i_uv 

 ~? 



provided the aeroplane is flying in a line parallel to the axis of x. 



Here again we notice that this relation is entirely independent of the 

 relative rates of the central clocks at O and 0'. 



This relation between u' and u can be expressed much more compactly 

 by the use of hyperbolic functions. Thus, if we determine U, U', and V 

 so that 



- =tanh U, — =tanh IT, and - =tanh V, 

 c ' c ' c ' 



then the relation in question becomes simply 



U'=U-V. 



18. Finally, suppose the aviator flying over the platform carries a clock, 

 the observed rate of which with respect to S is R, and let us inquire what 

 the observed rate of the clock will be with respect to S. 



