128 PROCEEDINGS OF THE AMERICAN ACADEMY 



separated by au interval of 12 hours. By employing a large number 

 of stars, acciilentiil errors may be nearly eliminated from the mean of 

 any group. With this standard catalogue any other eatnlogue may be 

 compared, and from the mean of the residuals about each hour of Kight 

 Ascension there may be formed 2-i equations of the form : — 



m sin l"" -I- n cos P = r, 



m sin 2'' + n cos 2''= r^ &c., 



from which ?» and n can be obtained by the process of least-squares. 

 Either the Pulkowa c;xtalo2;ue for 1845, the Aboecataloaue for 1828, or 

 the Dorpat catalogue for 1830, might be selected for this purpose with 

 great advantage, were it not for the occasional uncertainty in the value 

 of the proper motions with which the Kight Ascensions are brouglit for- 

 ward to the present time. But it will be seen from this investigation, 

 that no series of observations made since 1858 is wholly free from the 

 errors in question. 



In the earlier investigations on this subject, especially in the one 

 given by Professor SafFord (Monthly Notices of the Royal Astronomi- 

 cal Society, Vol. XXI., No. 9), who seems to have been the first to 

 suggest the present form of discussion, the standard or Greenwich 12 

 Year Catalogue, is assumed to be free from periodic errors. A similar 

 assumption with respect to the Washington Observations seems "also 

 to have been ptu'tially made by Professor Newcomb, in deriving the 

 periodic equation for Dr. Gould's catalogue of fundamental stars from 

 the Washington Observations from 1862 to 1867. (Washington Obser- 

 vations for 1867.) Wliile the error arising from this source is incon- 

 siderable in the latter case, it has nevertheless a sensible magnitude. 



II. Instead of depending upon a large number of stars to secure free- 

 dom from accidental errors, we may employ only the Maskelyne fun- 

 damental stars of the lirst and second magnitude, reiving for our 

 purpose, upon their more accurately determined places. If the assumed 

 Eight Ascensions of these stars have been determined by a process from 

 which ditterential observations have been excluded, and in which great 

 care has been taken to employ only data from which periodic errors 

 have been eliminated bv the method of observation, the whole svstem 

 of Eight Ascensions may be regarded as homogeneous. Even if this 

 condition is not exactly fidtilled, if the periodic coefficients for ditlerent 

 catalogues have opposite signs, the resulting system may safely be as- 

 sumed to be nearly homogeneous, and nearly free from periodic errors 

 of all kinds. 



