64 PBOCEEDINGS OF THE AMERICAN ACADEMY 



The mean sine of ^ is about equal to the integral 



divided by the integral 



^ sm2x5x 





X 5 X 



or unity ; hence equal to i tt = 0.7854. 



The value of ■ ^ - will then, from the present series, be roughly 



r A s 



equal to 0.556 -~ 0.785 = 0.708, 



with a probable error of about ± 0.023. 



But it will be noticed that in the present group of stars the values of 

 cos. {\\)' — I/') increase as the proper motions become smaller ; in the 

 former paper they appeared to decrease; making it still more probable 

 that the variations in both series arise from the errors of observation, 

 and the very various regions in which 'the stars are situated, and not 

 from any general deviation from constancy in the ratio 



r A s 



The value of this ratio, from the previous series, was found equal to 

 0.666 for forty -three- groups of the largest proper motions, and to 0.46 

 for eleven groups of smaller, or in the mean about 0.622 ; the best 

 of the two values agreeing nearly with the present determination. 



This is sufficiently noteworthy, as the stars of the former series 

 were, on the whole, those visible to the naked eye, or of the magni- 

 tudes one to six inclusive ; of the present ones, very few are brighter 

 than the sixth, and many are of the eighth and ninth ; so that so far 

 the phenomena indicate hardly any dependence upon magnitude. In 

 a future paper, I shall discuss a considerable number of other notable 

 proper motions, which I myself have been lately employed in deter- 

 mining for a catalogue of latitude stars ; and I intend to gradually 

 accumulate materials for a more minute investigation of the whole 

 subject, separating the stars according to their magnitudes and the 

 region of the heavens in which they are situated. 



The delicacy of the investigation is very great, owing to the danger 

 of taking that for stellar motion which is simply the result of the 

 errors of observation. Of course the old observers had indifferent 

 instruments ; but as a rule they used them well, if their work had plan 

 and coherency enough to be preserved. Bradley's observations 

 (1755) are still very accurate and indispensable; Piazzi and Groom- 



