J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 333 



and between this latter limit and t = oo there remain only five 

 errors ; a number so small that it is scarcely probable that any- 

 thing will ever be experimentally decided in respect to this de- 

 viation of the theory from the rule. 



Among the different values of t, there is one especially which 

 may lead to a determinate view as to the proportionate exact- 

 ness of different kinds of observations. It is that value of t for 

 which the integral has the value 0*5, or which, if we take a suf- 

 ficiently large number of errors, and imagine them arranged in 

 the order of their magnitudes without respect to signs, will di- 

 vide them into two parts, each containing an equal number of 

 errors. The number of errors is assumed to be large only in 

 order that the law of probability may actually be fulfilled with 

 sufficient approximation. From the table, it is found, by inter- 

 polation, that the value of the Integral 0'5 corresponds to the 

 value of ^ = 0-476936. Let this number, which holds good for 

 all kinds of observations, be designated by g, on account of the 

 frequent use to be made of it, so that 



§ = 0-476936 and ;^ / _ ^ ^' dt = ^. (9.) 



If we designate by r the error which belongs to the value 

 t = § in each kind of observations, then 



g= hr or A = -^. (10.) 



German astronomers call the magnitude ?- the probable error 

 of any particular class of observations*. It is that error below 

 which there are as many errors less than itself as there are 

 larger ones above it; so that there are as many cases in which 

 the errors are less than r, as there are cases in which the errors 

 are greater. Therefore it is an equal chance, that the error of an 



• The French geometricians are in the habit of giving to this value of r the 

 name of I'erreur moyenne : this is the more to be borne in mind, as the import 

 hitherto given by German writers to the term mean error differs essentially 

 from r. 



