Cc'ii WEIGHTS AND MEAN KRROR8. 



star from any one authority was rarely sufficient to render the resulting places essentially 

 different, whatever the principle adopted for the assignment of weights ; and secondly, so large 

 an amount of labor had already been expended on the determination of these places before my 

 attention was drawn to this question, that a repetition of the work would have entailed an 

 expenditure of time and care altogether disproportionate to the possible increase of accuracy in 

 the resultant places ; an increase which would certainly have been of a different order of mag- 

 nitude from the inevitable uncertainly of the ultimate determination. The numbers annexed to 

 the star-places of our General Catalogue, and there denominated weights, should in truth be 

 divided by a factor which for the majority of cases is not far from constant ; but the assump- 

 tion of a limit of accuracy beyond which the multiplication of observations is comparatively 

 useless renders even such division much less important, and these quantities may therefore be 

 practically regarded as representing the number of standard observations from which the 

 adopted positions are derived. This number is generally large enough to render the assump- 

 tion very reasonable, and the principle has been followed throughout the present computation. 



For the determination of the value of a measured position from the number of its constituent 

 observations, or of the weight of an observation as a function of the number of comparisons 

 upon Which it depends, let us, as the simplest, and a fully sufficient means of attaining the end 

 designed, consider each of the set of individual comparisons as affected with a constant error c, 

 and introduce this constant error as a multiple of the theoretical mean error e. For simplicity's 

 sake we will disregard the technical "probable error," to which, of course, the linear functions 

 of the mean error are convertible by multiplication with the constant factor, and will employ 

 only the quantity e, which we may call the probable discordance from the mean. 



We shall then have, by putting c= ae, the probable mean errors, thus : 



For the result from 1 observation, e. Va* -\- \ 



" " n observations, S.A a 9 +- 



n 



the weights of the two determinations being respectively TYH , and a , , Consequently 

 the result from n observations has, when compared with that from 1 observation, the weight 



which assumes a more convenient form if instead of a, the ratio of the errors, we introduce 

 b, the ratio of the weights, so that 



The determination of the value b is of necessity empirical, and its magnitude dependent upon 

 the quality of the observations, increasing in the ratio of their delicacy ; so that twenty-five 

 measurements from a source for which b := 5 are worth five times as much as one measurement, 

 thirty-six comparisons where 6 = 6 are worth six times as much as one, &c. No number of 

 measurements, however great, would, in the first instance, be six times, or in the second one, 

 seven times as valuable as a single one. 



Under some circumstances it may be found desirable to give another form to the expression 

 for the weight, especially when different sets of observations are to be compared with one 



