J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 359 



and 



1.2.3...2n 2'^ " 



(1.2. 3. ..«)'- ^n^r' 



EuLER Calc. Diff. P. ii. Cap. vi. § 160-162, as the limiting value 

 to which it continually approximates as n increases ; so that the 

 expression becomes 



2 » + 1 f^^Vn^ (0 



€ * ds; 



n V 



«y 



for which we need not scruple to write 



e-'' ds, 



— f 



ty 



as the expression of the probability that with numerous ob- 

 servations, the middlemost error, all being arranged accord- 

 ing to then' magnitudes, lies between r — 8 and ?• + S. This 

 probability consequently becomes ^, or the probable limits are 

 given by 



2^ V nyfr (r) = g, 

 whence 



8 = 



For the law of the errors assumed above 



the probable limits of r will be 



4 y' » . A 



or if, instead of 2 « + 1, we call the number of observations m, 

 and if we employ the equation hr = q, 





v^ (8 m) 



The numerical value of e«^ is 1*2554176, whereby the expres- 

 sion becomes 



^{^±^^^}- 



This mode of determination of r is consequently still more inexact 



