198 PROCEEDINGS OF THE AMERICAN ACADEMY 



Great assistance has been obtained by noting all the doubtful cases, 

 and requesting their re-observation at the United States Naval Obser- 

 vatory, or re-observing them myself, when other duties would permit ; 

 but this is a matter which requires time. I might here mention that 

 the future progress of sidereal astronomy proper will be greatly fur- 

 thered by continually watching stars of doubtful proper motion until 

 their motions are decided ; and that the next few years will very greatly 

 increase the necessity of so doing. 



It is, therefore, very necessary that the system of co-operation among 

 the astronomers of different countries, which has lately begun, should 

 extend much wider than it has, or at least that every observer should 

 strive to regulate his work by the uniform principles which guide the 

 best ones, and also to do that which is most necessary for the general 

 good. Fortunately, the making good observations, and reducing them 

 well, requires chiefly industry, system, and order, and is not very 

 dependent upon the capacity to appreciate the highest flights of mathe- 

 matical genins. 



The one thing needful for good observations is, in a word, disci- 

 pline. 



In computing probable errors, I generally use the sums of the dis- 

 crepancies, not the sums of their squares. The formula is. — 



£ = 0.840 ./— 7 X 



^ m{m — n) 



where ^e denotes the sum of the errors, £ the probable error of a 

 single one, jn their number, ?^ the number of unknown quantities. 

 The little table annexed contains 



V'w (m — n) 



31 = 



0.845 



so that 



with the arguments m and n. I have extended it only so far as I 

 habitually use it ; beyond these limits, it is better to calculate by loga- 

 rithms. 



In the cah'ulations above-mentioned, I have used two little devices 

 for shortening the solution by least squares, which are best illustrated 

 by the same example. 



