RECENT ADVANCES IN SCIENCE 559 



so contrary to " common sense." But it should be noted that 

 the statement is about two events in different places for which 

 the only test of simultaneity or non-simultaneity is the agree- 

 ment or disagreement of time-marks recorded on different clocks 

 at the instants of occurrence. There is nothing in this to con- 

 tradict any impressions made in the mind of a single observer 

 as to the time relations to him of two events that come under 

 his own direct observation. 



It is easily derived from the above equations that if a body 

 (idealised as a particle) moves from a point P (xyz in the S-axes, 

 x'y'z' in the S' axes) to a point Q (x + dx, y + dy, z rf- dz 

 in the S axes, x' + dx' , etc., in the S') between one instant 

 (/ for S, f for S') and another (t + di for S, t' + dt' for S'), 

 then dx, dy, dz and dt are connected with dx' , dy' , and dt' , by 

 the very same equations as before, and hence 



dP - dx 2 - dy 2 - dz 2 = dt' 2 - dx' 2 - dy' 2 - dz' 2 . (2) 



Therefore dt.^J 1 — v 2 = dt'.s/i — v' 2 , 

 where v and v' are the velocities of the body relative to S and 

 S' respectively. If there were observers on the body itself 

 so that the body would be fixed in their frame of reference 

 (and for them v = o), their estimate of the interval would be 

 dr, where dr = dty/i - v 2 (dl/0) = dt'^/i - v' 2 (dt//3'). This is 

 an element of what is called " proper " time for this body. 

 To observers on the body supplied with the spatial measure- 

 ments of S but adopting the " proper " time of themselves, 

 the velocity would be dx/dr, dy/dr, dz/dr and acceleration 

 would be d 2 x/dr 2 , d 2 y/dr 2 , d 2 z/dr 2 . Let us write down as a 

 tentative law of motion 



Hd 2 x/dr 2 = P x ) 



vd 2 y/dr 2 =P y \ (3) 



^d 2 z 2 /dr 2 = Pj 



P x , P y , P z being components of a vector and u being a fixed 

 number for the body. If we now turn to the S' observers, 

 they would write the above law as 



//. d 2 x' /dr 2 = P z ,, and non-similar" 



where P x , = a(P x — u //, -J-\ 



P = P 



p.= p 



(4) 



Translated from " proper " time to times in the system S, 

 we would have 



dA^T/?) ~ ^ x anc * two s ^ m ^ ar (5) 



