\magnification=\magstep2


$$
{\rm Fred\ measures\ } (x, y, z, t)   \quad  \Rightarrow \quad
       {\rm Jane\ measures\ } (x', y', z', t') 
$$

$$
x, y, z, t \quad {\rm \ and\ relative\ speed\ } \quad \phantom{-}v \quad {\rm \ along\ x\ axis\ }
$$

$$
x', y', z', t' \quad {\rm \ and\ relative\ speed\ } \quad -v \quad {\rm \ along\ x\ axis\ }
$$

$$
\eqalign{
x' &= x - vt  \cr
y' &= y   \cr
z' &= z  \cr
t' &= t   \cr
}
$$


$$
\eqalign{
x' &= \gamma( x - vt )  \cr
y' &= y   \cr
z' &= z  \cr
t' &= \gamma(t - {{vx}\over{c^2}})   \cr
}
$$

$$
t' = \gamma(t - vx) 
$$


$$
v' = v - (- w) = v + w
$$

$$
\eqalign{
{\rm TWEET\  } \qquad &(x_1, t_1) \cr
{\rm meet\ dog \  } \qquad &(x_2, t_2) \cr
}
$$

$$
\eqalign{
{\rm TWEET\  } \qquad &x'_1 = x_1 - (-w) t_1 \quad, \quad t'_1 = t_1 \cr
{\rm meet\ dog \  } \qquad &x'_2 = x_2 - (-w) t_2 \quad, \quad t'_2 = t_2 \cr
}
$$

\bigskip

$$
\eqalign{
v' &= {{(x'_2 - x'_1)}\over {(t'_2 - t'_1)} } \cr
\cr
  &= {{(x_2 - (-w) t_2) - (x_1 - (-w) t_1) } \over 
        {(t_2 - t_1) } } \cr
\cr
   &=  {{(x_2 - x_1)} \over {(t_2 - t_1) } } + 
                {{w (t_2 - t_1)} \over {(t_2 - t_1)}} \cr
\cr
   &=   v + w  \qquad\quad {\rm (Galilean\ version)}\cr
}
$$


$$
\eqalign{
{\rm TWEET\  } \qquad x'_1 &= \gamma(x_1 - (-w) t_1) \cr
               t'_1 &= \gamma\left ( t_1 - {{(-w) x_1}\over{c^2}} \right ) \cr
{\rm meet\ dog \  } \qquad x'_2 &= \gamma(x_2 - (-w) t_2) \cr
               t'_2 &= \gamma\left ( t_2 - {{(-w) x_2}\over{c^2}} \right ) \cr
}
$$


$$
\eqalign{
v' &= \quad {{(x'_2 - x'_1)}\over {(t'_2 - t'_1)} } \cr
\cr
  &=  \quad { {\gamma{(x_2 - (-w) t_2) - \gamma(x_1 - (-w) t_1) } } \over 
     {   \gamma \left ( t_2 - {{(-w) x_2}\over{c^2}} \right ) -
         \gamma \left ( t_1 - {{(-w) x_1}\over{c^2}} \right ) } } \cr
\cr
  &=  \quad { {x_2 + w t_2  - x_1 - w t_1  } \over 
     {    t_2 + {{w x_2}\over{c^2}} - 
          t_1 - {{w x_1}\over{c^2}} }   } \cr
\cr
  &=  \quad { {(x_2 - x_1) + w (t_2 - t_1)  } \over 
     {    (t_2  - t_1)  + {w {(x_2 - x_1)}\over{c^2}}  }  }   \cr
}
$$

\bigskip

$$
v' \quad = \quad { {v + w} \over { 1 + {{vw}\over{c^2}} } }
$$



\bye
