Page:CarmichealPostulates.djvu/21

in terms of t', x', y', z'. But these can be found more easily by solving equations (A) for t', x', y', z'. Thus we have

$(A_{1})$

${\begin{array}{llc}t=&{\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t'+{\frac {v}{c^{2}}}x'\right),\\x=&{\frac {1}{\sqrt {1-\beta ^{2}}}}(x'+vt'),&(MVLR)\\y=&y'\\z=&z',\end{array}}$

These two sets of equations (A) and ($A_{1}$) are identical in form except for the sign of v. This symmetry in the transformations constitutes one of their chief points of interest.

Our method of proof of these formulae is very different from that of Einstein, as a comparison will readily show. The difference is due primarily to our use of postulates V and L instead of the assumptions of Einstein.^[1]

In a paper^[2] entitled "The Common Sense of Relativity" Campbell has made some interesting remarks concerning these transformations.

§ II. The Addition of Velocities. — For the sake of completeness^[3] in the presentation of the fundamental results of relativity and for use in the next section we derive here the formulae for addition of velocities due to Einstein.^[4]

Let the velocity of a point in motion be represented in units belonging to $S'$ and to $S$ by means of the equations

${\begin{array}{lll}x'=u_{x'}t,&y'=u_{y'}t,&z'=u_{z'}t,\\x=u_{x}t,&y=u_{y}t,&z=u_{z}t,\end{array}}$

respectively. In the first of these substitute for $t',x',y',z'$ their values given by (A), solve for x/t, y/t, z/t and replace these quantities by their equals $u_{x},u_{y},u_{z}$ respectively. Thus we have

(B)	${\begin{array}{lc}u_{x}={\frac {u_{x'}+v}{1+{\frac {vu_{x'}}{c^{2}}}}}\\\\u_{y}={\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {vu_{x'}}{c^{2}}}}}u_{y'},&(MVLR)\\\\u_{z}={\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {vu_{x'}}{c^{2}}}}}u_{z'}\end{array}}$

1. Einstein, l. c. p. 420, footnote.
2. Phil. Mag.. 21 (1911): 502-517; see esp. pp. 505-507.
3. See remarks in the Introduction.
4. Einstein, l. c, pp. 422-424.