334 H. Jacoby — Determination of the Division 



This method of Hansen appears to have received no mate- 

 rial improvement until 1888. In that year, Gill, acting upon a 

 suggestion made by Marth, began to determine the errors of 

 the straight scales of the Cape heliometer. Gill employs an 

 auxiliary scale B which is an exact duplicate of scale A. In 

 fact, in the case of the heliometer, each of the two scales is 

 used as an auxiliary for investigating the other. Instead of 

 then comparing, as Hansen did, every one-space, two-space, 

 etc., of scale A with one auxiliary one-space, two-space, etc., on 

 scale B, Gill compares every one-space of scale A with every 

 one-space of scale B ; every two-space with every two-space ; 

 etc. This method must be regarded as an important extension 

 of Hansen's. With comparatively little additional labor in 

 the observations, it causes a considerable increase of precision 

 in the results. 



Precisely the same method of observation employed by Gill 

 has been discussed very recently by Lorentzen,* and used in 

 determining the errors of the Bamberg heliometer. He begins 

 by applying the method of least squares rigidly to the reduc- 

 tion of the observations ; but the numerical work required 

 being very great, he concludes by recommending a slight depar- 

 ture from the rigid least square method of reduction. This 

 brings him to a method which is identical with Gill's, both in 

 the observations, and in the calculation of the division errors. 

 They differ, however, in the calculation of the weights of the 

 division errors determined for the several lines of the scale. 

 Gill comes to the conclusion that all these division errors are 

 determined with the same weight. This is not rigorously 

 correct. Lorentzen does not come to the same conclusion ; but 

 as we shall see further on, he deduces a weight formula which 

 is quite correct. 



2. The method of Hansen, both in its original form, and as 

 extended by Gill, in common with all other methods of deter- 

 mining division errors, has the defect just mentioned. The 

 various lines are not determined with rigorously equal weight. 

 Since, in general, there is no reason a priori why we should 

 determine some of the lines more accurately than the others, a 

 method which will produce true equality of weights for all the 

 lines still remains a desideratum. Now the method of Gill 

 would appear to be observationally exhaustive, since all possi- 

 ble combinations of spaces on the two scales have been com- 

 pared. Nevertheless, the method may be observationally 

 varied, the general weight greatly increased, and even the 

 desired equality of weights may be produced, if we vary the 

 precision with which the several comparisons between the two 

 scales are effected. This need not increase the labor very 



* Astronomische Nachrichten, 3134, 3236. 



