454 KRETZ. 



[A] ^ [Al [A] 



[A] ^ + [A] [A] ' 



and 



iP,X,X, . l^p' + [AA'i F, . i]r^ + [AA>x • I] =o 

 [AFiFj. !];-/+ [/,Fi«.. I] =o 



[A J^2 ^2 •!]/'+ [ A^l, F, . l\r' + [A n% • I] = o 

 Ut^i^t ■ I]'-' + [AA'2% ■ I] =0 



Add the first and third, and the second and fourth of the last 

 equations, term to term, and from the resulting equations ob- 

 tain /' and /. The values will be identically the same as if 

 all the four unknowns had been eliminated from one set of 

 normals by the general method. 



The weights of the unknowns could, in this case, at once be 

 written down, with sufficient accuracy. For owing to the fact 

 that the weights in right ascension and in declination of the 

 observation equations are nearly equal, we have 



\_p,X,X, ■ I] + [A Y, F, . I] = [A F, F, - I] + \_p,X,X, . I] 



nearly and 



[AATiFi-iJ + CAA-^F,-!] 



small, so that we can place (cf Chauvenet, Astronomy, Vol. II, 

 P- 537) 



Wt of /^ = lp,X,X, . ij + [A F, F, . I] 

 Wt. of r^=[/FF- 3] 



where \_pYY- 3] denotes the coefficient of r' in the last 

 elimination equation. Similarly, in the inverted elimination, the 

 coefficients of /' and r' are very large compared to those of 

 k and c, so Inat at once 



Wt. k = [A] of the equations containing k 

 Wt. <r = [A] of the equations containing c. 



Knowing the weights, the probable errors were then obtained 

 in the usual manner from the residuals. 



(114) 



