558 SCIENCE PROGRESS 



given by time and space co-ordinates t' , %' , y' , z' , their axes 

 being fixed relative to them. The equations connecting 

 these eight quantities are a set of four linear equations. They 

 take a simple and sufficiently general form if the axes OX and 

 O'X' have been chosen by the observers to be parallel to their 

 line of relative motion and the origins and 0' to have coin- 

 cided at the time t = o for the S-observers, and t' = o for 

 the S'-observers. 



They become, in that case, 



x' = a (x — id) x = a {x' — u't'Y 



y' =y y, = y 



z' — z z = z' 



t' = a (t — itn) t = (f — a'x'). 



(0 



In these, we have assumed the units of length and time 

 to be chosen so that the unit of velocity is the velocity of 

 light ; the relative velocity of S' to S, as measured by S, is u 

 parallel to OX, and of S to S', as measured by S', is n' parallel 

 to OX', so that u' = — u (the symbol u is, in reality, the 

 ratio of the velocity in question to the velocity of light). The 

 quantity a = \ /^/i —u 2 — i/V 1 — w /a . 



Three consequences follow from these equations : 

 (i) The dimensions of a body are not the same to observers 

 for whom it is fixed as to observers for whom it is in motion. 

 If the first group measure a length in it to be /, the second 

 group measure it to be l/a or i\/i —u 2 , if it is parallel to the 

 relative motion, and to be / (i.e. the same) if perpendicular. 

 That is, bodies are shorter to observers past whom they are 

 moving, in a direction parallel to the motion. (The contrac- 

 tion is, of course, excessively small for usual velocities.) 



(2) Two events occur at a locality. The interval between 

 them is measured as t units of time by observers fixed i n this 

 locality. This interval will be measured as a t or t/<\/i—u 2 

 units by observers in relative motion to the locality. That 

 is, all periodic mechanisms appear to go slower to observers 

 in relative motion to them than to observers at rest relative 

 to them. (It is implied, of course, that the mechanisms used 

 by S and S' to measure time would synchronise perfectly if 

 all were at rest to one another, and under direct observation 

 by any one observer.) 



(3) An event which occurs at one place is regarded as 

 simultaneous with another event which occurs at another 

 place by a group of observers. The events will not be regarded 

 as simultaneous by any other group of observers who are in 

 relative motion to the first group. 



It is this last conclusion which appears to the lay mind as 



