106 Reduction of Stellar Photographs. 



Let us, for brevity, indicate the coefficients of p and r in the 

 right ascension equations by r. and p. Let us also represent by v 

 the number of stars, so that there will be v right ascension equa- 

 tions, and as many declination equations. The general form of 

 the equations will then be : 



From the right ascensions : 



vrp -\- pr -\- k ^ n'x =0 

 where, for brevity, we have written ??^' for i^n^ cos d. 

 From the declinations : 



PP — TT r + c + % =0 

 If we now indicate by square brackets the summation of v quan- 

 tities, the rigorous least square solution of the above 2v equa- 

 tions is given by the following simple sj-stem of formulae : 



c+c 



P = —A+D' weight of p=A -j- D 



E+E' 



A+D^ 



weight of r^A-\-D 



^ = -7 { [^] P+iP^ r+ [n'.j } , weight of h=v-.^^±^^ 

 C = -^{Mi>-Mr+K] j-,weightof c=v-^^^^ 



By the aid of these formulas, the rigorous least square solution 

 can be made in about half an hour, including the determination 

 of the weights. 



The total corrections to be added to the co-ordinates of the un- 

 known stars will then be : 



{p+M:c) Xsec rf+Ci^^ r sec S+N^) F+ ^V h sec i\ for X sec rf | 

 {— IS r cos S+3Iy)X sec 6+{p-\-N,j)Y+c, " Y j 



which are very readil}'^ computed with Crelle's tables. After this to- 



