168 PROCEEDINGS OP THE AMERICAN ACADEMY 



XIII. 



ON THE LIMITS OF ACCURACY IN MEASUREMENTS 

 WITH THE TELESCOPE AND THE MICROSCOPE. 



By Professor William A. Rogers. 



Presented Oct. 9, 1878. 



It is often desirable in astronomical observations to assign to a 

 given result the degree of precision which the observations will 

 justify. Usually the limit of precision is defined either by the proba- 

 ble error of a single observation, or of the mean of a given number of 

 observations. 

 Let 



x = any given numerical value. 

 n = the number of values of x. 



v = the difference between each value of x and the 

 arithmetical mean of all the values, 

 [v] = the sum of the separate residuals, without regard 

 to sign. 

 r = the probable error of a single value. 

 r = the probable error of the arithmetical mean. 



We shall then have, — 



r = .674 5 y/S 

 Or, according to Peters, — 



(«) 



r = .8453 - 

 r n = .8453 



tfm (m — 1) 



(*) 



