\magnification=\magstep1

$$
L \quad = \quad L_0 \ + \ 0.9856474 (t - t_0)
$$

$$
g \quad = \quad g_0 \ + \ 0.9856003 (t - t_0)
$$

$$
\lambda \quad = \quad L \ + \ 1.915 \sin(g) \ + \ 0.020 \sin(2 g)
$$


$$
(\alpha - \alpha_0) \quad = \quad
  \varpi \bigl[ \cos(\alpha) \cos(\epsilon) \sin(\lambda)
                    \ -  \ \sin(\alpha) \cos(\lambda) \bigr]
   \ + \ \mu_\alpha t 
   \ + \ K
$$

$$
\cos(\alpha_0) \quad \simeq \quad \cos(\alpha)
$$

$$
\sin(\alpha_0) \quad \simeq \quad \sin(\alpha)
$$

$$
\cos(\delta_0) \quad \simeq \quad \cos(\delta)
$$

$$
\sin(\delta_0) \quad \simeq \quad \sin(\delta)
$$

$$
(\alpha - \alpha_0) \quad \simeq \quad
  \varpi \bigl[ \cos(\alpha_0) \cos(\epsilon) \sin(\lambda)
                    \ -  \ \sin(\alpha_0) \cos(\lambda) \bigr]
   \ + \ \mu_\alpha t 
   \ + \ K
$$

$$
P \quad \equiv \quad \cos(\alpha_0) \cos(\epsilon) \sin(\lambda)
                  \ - \ \sin(\alpha_0) \cos(\lambda)
$$

$$
(\alpha - \alpha_0) \quad \simeq \quad
  \varpi P
   \ + \ \mu_\alpha t 
   \ + \ K
$$

$$
\eqalign{
(\delta - \delta_0) \quad = \quad 
         \varpi \bigl[ \cos(\delta) \sin(\epsilon) \sin(\lambda)
            & \ - \ \cos(\alpha) \sin(\delta) \cos(\lambda) \cr
            & \ - \ \sin(\alpha) \sin(\delta) \cos(\epsilon) \sin(\lambda) \bigr]
   \ + \ \mu_\delta t
   \ + \ L \cr
}
$$


$$
\eqalign{
(\delta - \delta_0) \quad = \quad 
         \varpi \bigl[ \cos(\delta_0) \sin(\epsilon) \sin(\lambda)
            & \ - \ \cos(\alpha_0) \sin(\delta_0) \cos(\lambda) \cr
            & \ - \ \sin(\alpha_0) \sin(\delta_0) \cos(\epsilon) \sin(\lambda) \bigr]
   \ + \ \mu_\delta t
   \ + \ L \cr
}
$$

$$
\eqalign{
Q \quad \equiv \quad 
          \cos(\delta_0) \sin(\epsilon) \sin(\lambda)
            & \ - \ \cos(\alpha_0) \sin(\delta_0) \cos(\lambda) \cr
            & \ - \ \sin(\alpha_0) \sin(\delta_0) \cos(\epsilon) \sin(\lambda)  \cr
}
$$



$$
(\delta - \delta_0) \quad = \quad \varpi Q \ + \ \mu_\delta t \ + \ L 
$$

$$
\beta \quad = \quad \sum {
       \  
           {\bigl[ \Delta {\rm RA}_i - (\alpha - \alpha_0) \bigr]}^2 \ + \ 
           {\bigl[ \Delta {\rm Dec}_i - (\delta - \delta_0) \bigr]}^2 
  }
$$

$$
\beta \quad = \quad \sum {
       \  
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]}^2 \ + \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]}^2 
  }
$$

$$
\eqalign{
{{\partial \beta}\over{\partial \varpi}} \quad = \quad \sum 
       \ 
          &(-2 P_i) \ 
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]} \ +  \cr
          &(-2 Q_i) \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]} \cr
}
$$

$$
{{\partial \beta}\over{\partial \varpi}} \quad = \quad 0
$$


$$
\eqalign{
0 \quad = \quad 
          &\sum  \Delta {\rm RA}_i P_i \ - \ 
           \sum  \varpi P_i^2 \ - \  
           \sum  \mu_\alpha P_i t_i \ - \ 
           \sum  K P_i  \ + \ \cr
          &\sum  \Delta {\rm Dec}_i Q_i \ - \ 
           \sum  \varpi Q_i^2 \ - \  
           \sum  \mu_\delta Q_i t_i \ - \ 
           \sum  L Q_i  \cr
}
$$

$$
\eqalign{
           \varpi \sum  \left( P_i^2 + Q_i^2 \right) \ + \  
           \mu_\alpha \sum   P_i t_i \ + \ 
           & \mu_\delta \sum Q_i t_i \ + \ 
           K \sum P_i  \ + \ 
           L \sum Q_i   \cr
   &\quad = \quad  
          \sum  \left( \Delta{\rm RA}_i P_i \ + \ \Delta {\rm Dec}_i Q_i \right)  \qquad\qquad\qquad(1)
}
$$


$$
{{\partial \beta}\over{\partial \mu_\alpha}} \quad = \quad
       \sum  {
          (-2 t_i) \ 
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]} 
        }
$$

$$
0 \quad = \quad 
           \sum  \Delta {\rm RA}_i t_i \ - \ 
           \sum  \varpi P_i t_i \ - \  
           \sum  \mu_\alpha t_i^2 \ - \ 
           \sum  K t_i  
$$

$$
           \varpi \sum  P_i t_i \ + \  
           \mu_\alpha \sum  t_i^2 \ + \ 
           K \sum  t_i  
     \quad = \quad 
           \sum  \Delta {\rm RA}_i t_i 
$$

$$
\eqno{
           \varpi \sum  P_i t_i \ + \  
           \mu_\alpha \sum  t_i^2 \ + \ 
           \mu_\delta \thinspace 0  \ + \ 
           K \sum  t_i  \ + \  
           L \thinspace 0  
     \quad = \quad 
           \sum  \Delta {\rm RA}_i t_i 
}{\qquad\qquad(2)}
$$


$$
{{\partial \beta}\over{\partial \mu_\delta}} \quad = \quad
       \sum {
          (-2 t_i) \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]} 
       }
$$

$$
0 \quad = \quad 
           \sum  \Delta {\rm Dec}_i t_i \ - \ 
           \sum  \varpi Q_i t_i \ - \  
           \sum  \mu_\delta t_i^2 \ - \ 
           \sum  L t_i  
$$


$$
           \varpi \sum  Q_i t_i \ + \  
           \mu_\delta \sum  t_i^2 \ + \ 
           L \sum  t_i  
     \quad = \quad 
           \sum  \Delta {\rm Dec}_i t_i 
$$


$$
\eqno{
           \varpi \sum  Q_i t_i \ + \  
           \mu_\alpha \thinspace 0 \ + \ 
           \mu_\delta \sum  t_i^2 \ + \ 
           K \thinspace 0  \ + \   
           L \sum  t_i  
     \quad = \quad 
           \sum  \Delta {\rm Dec}_i t_i    
}{\qquad\qquad(3)}
$$


     

$$
{{\partial \beta}\over{\partial K}} \quad = \quad
       \sum {
          (-2) \ 
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]} 
       }
$$

$$
0 \quad = \quad 
           \sum  \Delta {\rm RA}_i \ - \ 
           \sum  \varpi P_i \ - \  
           \sum  \mu_\alpha t_i \ - \ 
           \sum  K   
$$


$$
           \varpi \sum  P_i \ + \  
           \mu_\alpha \sum  t_i \ + \ 
           K \sum  1
     \quad = \quad 
           \sum  \Delta {\rm RA}_i  
$$

$$
\eqno{
           \varpi \sum  P_i \ + \  
           \mu_\alpha \sum  t_i \ + \ 
           \mu_\delta \thinspace 0 \ + \ 
           K \sum  1  \ + \ 
           L \thinspace 0
     \quad = \quad 
           \sum  \Delta {\rm RA}_i  
}{\qquad\qquad(4)}
$$

     
     

$$
{{\partial \beta}\over{\partial L}} \quad = \quad
       \sum {
          (-2) \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]} 
       }
$$

$$
0 \quad = \quad 
           \sum  \Delta {\rm Dec}_i \ - \ 
           \sum  \varpi Q_i \ - \  
           \sum  \mu_\delta t_i \ - \ 
           \sum  L   
$$


$$
           \varpi \sum  Q_i \ + \  
           \mu_\delta \sum  t_i \ + \ 
           L \sum  1
     \quad = \quad 
           \sum  \Delta {\rm Dec}_i  
$$

$$
\eqno{
           \varpi \sum  Q_i \ + \  
           \mu_\alpha \thinspace 0 \ + \ 
           \mu_\delta \sum  t_i \ + \ 
           K \thinspace 0 \ + \ 
           L \sum  1
     \quad = \quad 
           \sum  \Delta {\rm Dec}_i  
}{\qquad\qquad(5)}
$$

$$
\pmatrix{ \sum \left( P_i^2 + Q_i^2 \right) & \sum P_i t_i & 
          \sum Q_i t_i  & \sum P_i  &  \sum Q_i                \cr 
          \sum P_i t_i  & \sum t_i^2  &  0  &  \sum t_i  &  0  \cr 
          \sum Q_i t_i  &   0   &  \sum t_i^2   &  0  &  \sum t_i   \cr 
          \sum P_i      & \sum t_i  &  0   &  \sum 1  &  0     \cr
          \sum Q_i      &   0  &  \sum t_i &  0   & \sum 1     \cr  
}
  \ 
   \pmatrix { \varpi \cr  \mu_\alpha \cr  \mu_\delta \cr  K  \cr  L \cr }
      \quad = \quad 
   \pmatrix {  \sum \left( \Delta{\rm RA}_i P_i \ + \ \Delta{\rm Dec}_i Q_i \right)  \cr 
               \sum \Delta{\rm RA}_i t_i \cr 
               \sum \Delta{\rm Dec}_i t_i \cr 
               \sum \Delta{\rm RA}_i \cr 
               \sum \Delta{\rm Dec}_i \cr 
     }
$$


$$
{\bf A} \quad \equiv \quad \pmatrix{ \sum \left( P_i^2 + Q_i^2 \right) & \sum P_i t_i & 
          \sum Q_i t_i  & \sum P_i  &  \sum Q_i                \cr 
          \sum P_i t_i  & \sum t_i^2  &  0  &  \sum t_i  &  0  \cr 
          \sum Q_i t_i  &   0   &  \sum t_i^2   &  0  &  \sum t_i   \cr 
          \sum P_i      & \sum t_i  &  0   &  \sum 1  &  0     \cr
          \sum Q_i      &   0  &  \sum t_i &  0   & \sum 1     \cr  
}
$$

$$
{\bf x} \quad \equiv \quad 
   \pmatrix { \varpi \cr  \mu_\alpha \cr  \mu_\delta \cr  K  \cr  L \cr }
$$

$$
{\bf b} \quad \equiv \quad 
   \pmatrix {  \sum \left( \Delta{\rm RA}_i P_i \ + \ \Delta{\rm Dec}_i Q_i \right)  \cr 
               \sum \Delta{\rm RA}_i t_i \cr 
               \sum \Delta{\rm Dec}_i t_i \cr 
               \sum \Delta{\rm RA}_i \cr 
               \sum \Delta{\rm Dec}_i \cr 
     }
$$

$$
{\bf A x} \quad = \quad {\bf b}
$$

$$
\eqalign{
w_{{\rm RA},i} \quad &\equiv \quad {{1}\over{\sigma_{{\rm RA}_i}}} \cr
w_{{\rm Dec},i} \quad &\equiv \quad {{1}\over{\sigma_{{\rm Dec}_i}}} \cr
}
$$

$$
\beta \quad = \quad \sum {
       \  
 {\bigl[ \Delta {\rm RA}_i - (\alpha - \alpha_0) \bigr]}^2 w_{{\rm RA},i}^2 \ + \ 
 {\bigl[ \Delta {\rm Dec}_i - (\delta - \delta_0) \bigr]}^2 w_{{\rm Dec},i}^2 
  }
$$

$$
\eqalign{
\beta \quad = \quad \sum  \  &{\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]}^2 w_{{\rm RA},i}^2 \ +   \cr
           &{\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]}^2 w_{{\rm Dec},i}^2  \cr 
}
$$

$$
\eqalign{
{{\partial \beta}\over{\partial \varpi}} \quad = \quad \sum 
       \ 
          &(-2 P_i) \ 
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]} \thinspace w_{{\rm RA},i}^2 \ +  \cr
          &(-2 Q_i) \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]}  \thinspace w_{{\rm Dec},i}^2 \cr
}
$$

$$
\eqalign{
           \varpi \sum  \left( P_i^2 \thinspace w_{{\rm RA},i}^2 + 
                               Q_i^2 \thinspace w_{{\rm Dec},i}^2 \right) \ + \ 
           &\mu_\alpha \sum   P_i t_i \thinspace w_{{\rm RA},i}^2 \ + \  \cr
            \mu_\delta \sum Q_i t_i \thinspace w_{{\rm Dec},i}^2 \ + \  
           &K \sum P_i  \thinspace w_{{\rm RA},i}^2 \ + \ 
           L \sum Q_i  \thinspace w_{{\rm Dec},i}^2  \cr
   \quad &= \quad  
          \sum  \left( \Delta{\rm RA}_i P_i \thinspace w_{{\rm RA},i}^2 \ + \ 
                       \Delta {\rm Dec}_i Q_i \thinspace w_{{\rm Dec},i}^2 \right)  \cr
}
$$



$$
{\bf A} \quad \equiv \quad \pmatrix{ 
       \sum \left( P_i^2 \thinspace w_{{\rm RA},i}^2 + 
                   Q_i^2 \thinspace w_{{\rm Dec},i}^2 \right) & 
          \sum P_i t_i \thinspace w_{{\rm RA},i}^2 & 
          \sum Q_i t_i \thinspace w_{{\rm Dec},i}^2  & 
          \sum P_i \thinspace w_{{\rm RA},i}^2 &  
          \sum Q_i \thinspace w_{{\rm Dec},i}^2 \cr 
       \sum P_i t_i \thinspace w_{{\rm RA},i}^2 & 
          \sum t_i^2 \thinspace w_{{\rm RA},i}^2  &  
          0  &  
          \sum t_i \thinspace w_{{\rm RA},i}^2 &  
          0  \cr 
       \sum Q_i t_i \thinspace w_{{\rm Dec},i}^2 &   
          0   &  
          \sum t_i^2 \thinspace w_{{\rm Dec},i}^2  &  
          0  &  
          \sum t_i \thinspace w_{{\rm Dec},i}^2  \cr 
       \sum P_i \thinspace w_{{\rm RA},i}^2      & 
          \sum t_i \thinspace w_{{\rm RA},i}^2 &  
          0   &  
          \sum \thinspace w_{{\rm RA},i}^2 &  
          0     \cr
       \sum Q_i \thinspace w_{{\rm Dec},i}^2     &   
          0  &  
          \sum t_i \thinspace w_{{\rm Dec},i}^2 &  
          0   & 
          \sum \thinspace w_{{\rm Dec},i}^2    \cr  
}
$$

$$
{\bf x} \quad \equiv \quad 
   \pmatrix { \varpi \cr  \mu\alpha \cr  \mu_\delta \cr  K  \cr  L \cr }
$$

$$
{\bf b} \quad \equiv \quad 
   \pmatrix {  \sum \left( \Delta{\rm RA}_i P_i \thinspace w_{{\rm RA},i}^2 \ + \ 
                           \Delta{\rm Dec}_i Q_i \thinspace w_{{\rm Dec},i}^2 \right)  \cr 
               \sum \Delta{\rm RA}_i t_i \thinspace w_{{\rm RA},i}^2 \cr 
               \sum \Delta{\rm Dec}_i t_i \thinspace w_{{\rm Dec},i}^2 \cr 
               \sum \Delta{\rm RA}_i \thinspace w_{{\rm RA},i}^2 \cr 
               \sum \Delta{\rm Dec}_i \thinspace w_{{\rm Dec},i}^2 \cr 
     }
$$

$$
{\bf A x} \quad = \quad {\bf b}
$$



$$
\beta \quad = \quad \sum {
       \  
           {\bigl[ { {\Delta {\rm RA}_i - (\alpha - \alpha_0)}\over{\sigma_{{\rm RA}_i}} } \bigr]}^2 \ + \ 
           {\bigl[ { {\Delta {\rm Dec}_i - (\delta - \delta_0)}\over{\sigma_{{\rm Dec}_i}} } \bigr]}^2 
  }
$$



$$
\beta \quad = \quad \sum {
       \  
           {\bigl[ \Delta {\rm RA}_i - (\varpi P_i - \mu_\alpha t_i - K) \bigr]}^2 \ + \ 
           {\bigl[ \Delta {\rm Dec}_i - (\varpi Q_i - \mu_\delta t_i - L) \bigr]}^2 
  }
$$


$$
{{\partial \beta}\over{\partial \varpi}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial \varpi}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial \varpi}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial \mu_{\alpha}}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial K}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial \mu_{\delta}}} \quad = \quad 0 
$$

$$
{{\partial \beta}\over{\partial L}} \quad = \quad 0 
$$


$$
{\bf A x} \quad = \quad {\bf b}
$$

$$
{\bf x} \quad = \quad {\bf A^{-1}} \thinspace {\bf b}
$$


$$
{\rm PE} \quad = \quad a \ + \ bx \ + \ cx^2
$$


\bye
