﻿72 Dr. W. F. G. Swann on the Effects of Uniform Motion 



Now 



r a 2 = ?- 2 + 3f + 2r.Sf cos a, 



from which we obtain, on substituting for S£ the result given 

 by (6), the expression 



Cr V 2 C 2 OV 



Now consider the system at rest, and let M be the foot of 

 the perpendicular from P on the zy plane. If we make MQ 



i \ MP fi r2 V 12 



equal to £, we have, since — =- = I L— t^J, 



PQ = f[(l-J)"-l]; 



observing that the angle MQO is the same as the angle u in 

 the figure for the moving system, we see that if we write 

 OP-r,, 



,*= ,,+ P[(l -^)"- l] , +2rf [(l -c 3 )"-l] «»-. 



which reduces to 



- D-«t-i-)r 



Thus, since 0? = ^ and 0T"=rj, our expression for t' 

 becomes 



which after a little reduction reduces on writing «' for £ to 



e =/e v2 + — (7) 



Now write T for the time at which the instantaneous motion 

 considered occurs at in the system at rest ; the time at 

 which the corresponding motion occurs at 0' is e 1;2 T , so that 

 the time taken for the disturbance to be propagated to P' 



