\magnification=\magstep2


$$
\eqalign{
{\rm source\ \qquad \quad } & {\rm \qquad observer } \cr
t_0 = 0 {\rm \ yr}   \qquad \quad \quad & 0 {\rm \ yr\ } \quad + \quad\quad\quad  {{100 {\rm \ lyr\ }} \over {c} } \cr
\cr
t_1 = 1 {\rm \ yr}    \qquad \quad \quad & 1 {\rm \ yr\ } \quad + \quad  {{100 {\rm \ lyr\ } - (1 {\rm \ yr}) v \cos(\theta)} \over {c}} \cr
\cr
t_2 = 2 {\rm \ yr}    \qquad \quad \quad & 2 {\rm \ yr\ } \quad + \quad  {{100  {\rm \ lyr\ }- (2 {\rm \ yr}) v \cos(\theta)} \over {c}} \cr
}
$$

$$
\eqalign{
{\rm interval\ } \quad &= \quad 
   \quad\quad\quad t_1 \qquad\qquad\qquad\qquad\qquad -  \qquad t_0 \cr
\cr
 &= \left[ 1 {\rm \ yr\ } + {{100 {\rm \ lyr\ } - (1 {\rm \ yr}) v \cos(\theta)} \over {c}} \right]
        - \quad {{100 {\rm \ lyr\ }} \over {c} }  \cr
\cr
 &= \quad (1 {\rm \ yr }) \left[ 1 -  
              {  { v \cos(\theta)} \over {c} }  \right] \cr
}
$$

$$
{\rm distance \ } \quad = \quad v (1 {\rm \ yr}) \sin(\theta) 
$$

$$
\eqalign{
{\rm apparent\ speed\ } \quad &= \quad { {{\rm distance } \over {\rm interval }} } \cr
 \cr
       &= \quad {  { v (1 {\rm \ yr}) \sin(\theta) }  \over
             { (1 {\rm \ yr }) \left[ 1 -  {  { {(v/c)} \cos(\theta)} }  \right] } } \cr
\cr
       &= \quad {  { v \sin(\theta) } \over 
             { \left[ 1 -  {  { {(v/c)} \cos(\theta)} }  \right] } } \cr
}
$$

$$
{\rm apparent\ speed\ } \quad 
       = \quad {  { v \sin(\theta) } \over 
             { \left[ 1 -  {  { {(v/c)} \cos(\theta)} }  \right] } } 
$$

\bye
