J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 357 



If we regard the magnitude d t o( the interval between t and 

 t + dt a.s constant^ there is a value of t, for which U is a maxi- 

 mum. The equation obtained by differentiation for its deter- 

 mination is 



n±{fl_n±{{)^ 



u \ — u ' 



where \|/' t has the same signification as <$' A above, d u, or 



the increment of / \|/ A cf A, in reference to an infinitely small 



alteration of the limit t, is equal to '^ {t) d t. We may give to 

 the last equation the form 



u 1 — u n^) t 



The last member will be so much the less as n is greater, or 

 as there are more observations given. With a sufiiciently large 

 number it may be neglected. Or as n increases, the value of t, 

 for which the maximum takes place, approximates continually 

 to the value which follows from the equation 

 1 1 



:j = 0, 



whence 



u=J^^^{A)dA =1, 



or the value of r, according to the definition given above. 



If we take the integral of U between determinate limits, we 

 may obtain the probability that the error which is situated in the 

 middle is contained in these limits. It will be for the limits 

 r — 8 and r + 8 



1.2.3... (2n+ 1) f'' + ^ 

 = il.2.S.:.nr- j^ _ / (1 - ^)" ^ (^) ^^' 



or because "^ {t) dt ■=. d u, if for the Hmits in relation to / we 

 put 



then the probability that the middlemost value lies between r — S 

 and r -f 8 will be 



1 .2.3.... (2 w+ 1) /'"" „ , ,„ , 



= {1.2.3... nr J „, « (^ -^) ^«- 



