272 
PAUL BIEFELD 
in which k and k' are constants depending only on c, the velocity 
of light ; and v, the relative velocity of K' with respect to K. We 
calculate now from the second equation of (5) with the aid of 
the first of (5) : 
t'=k [t- 
-) 3 
( 6 ) 
and put the value of x', and t' from (5) and (6) into equation 
(4) A comparison of the coeficients of x\, t- and x^t leads to: 
k=k^ 
VI — — 
c2. 
Substituting k and k' in (5) and (6) we get: 
X^ vt t — ^ x^ 
xh=; , t' = , and x' 2 =Xo , x'^^x,. (7) 
VI— ^ Vl^"^ 
c2 C"- 
Solving for x^, t, x^ and x^ we get: 
x'l+vt' t'-f-^x'i 
Xi — , t = and Xo = x'., , X3 = x'. (8) 
Vl^'V 
These equations were for the first time developed by H. A. 
Lorentz hence the name '‘Lorentz transformation.’' 
Let us place a number of synchronous clocks in the system K. 
If their rates are zero they will remain synchronous. Some of 
these clocks are now transferred to the system K' having its axes 
parallel to those of K and moving relative to K with a uniform 
rate parallel to the X, axis. The clocks in K' as seen by an ob- 
server at rest in K, will depart from synchronism relative to the 
clocks in K, losing time or going slower. Rods of definite length 
as measured in K if placed in K' after they have been given the 
same motion as the system K', and placed in the X\ axis will 
seem contracted as viewed by the same observer at rest in K. If 
the velocity of K' with reference to K is known, both the change 
of rate of the clocks and the contraction of the rods may be nu- 
merically accounted for by means of the Lorentz transformation 
given above. It is of course evident that these changes will be 
extremely small on account of the second order effect: V^ oc- 
curring in the equation. If for instance v were as large as the 
orbital velocity of the earth, 18.5 miles per second, the value 
of the expression just given would be of the order 10.-® 
