WEIGHTS AND MEAN ERRORS. 



cell 



How-untrustworthy any function of this mean error would be as a guide in assigning the 

 proportionate weights to the different series before combining them in one, may he inferred from 

 a single glance at the values of x, approximately deduced from the different sets of observa 

 tions, on the assumption that the unknown quantities z and v are negligible. We may do this 

 by simply solving the equations : 



-*r 



and shall find- 



[aa] 

 [aa] 



[ab] [an] 

 =~WJ-' 



_ y 



[aa] 



The only method available for attaining impartial discrimination seems to be the determina- 

 tion of the mean error of an observation, by means of a preliminary solution of normal equations, 

 derived from empirical combination of the several groups for each planet-series. Assuming 

 the values thus obtained for our unknown quantities as sufficiently correct, we may substitute 

 them in the final equations of each group, and thus proceed to a determination of the weight 

 for each from the sum of the squares of the residuals outstanding after the substitution. 



At the same time we become entitled to compare the different values of the unknown quanti- 

 ties furnished by the several planet-series, and to remove the terms containing t, u, and v, which 

 will have been determined with sufficient approximation, and which have only been introduced 

 into our formulas for the sake of avoiding any possible error with which they might, if disre- 

 garded, affect our ultimate values for the remaining unknown quantities. 



For this preliminary solution, we may be permitted to attribute equal precision to the work 

 of the several observatories, omitting, however, all observations made at places for which the 

 details of the observations are inaccessible, and examination of the reductions consequently 

 impossible. The question of the relative value of the several groups can then be considered 

 with greater propriety. 



The relation of the weight of the observation to the number of comparisons of which it is 

 composed is the first topic for consideration. 



In forming the catalogue of star-places the ordinary principle of combination has been 

 retained, and for a two-fold reason. In the first place, the number of observations of the same 



