J. F. EXCKE ON THE METHOD OF LEAST SQUARES. 345 



Consequently, according to Proposition IV., the measure of pre- 

 cision for the value — of h 



1 , 



= Eg \/2 m or = -r- vm : 

 n 



and the probable error of this determination 

 qh _ q 1 ^ 



~ ^J m~ e^y/ 2 ' V m ' 

 or there is an even chance that the true value of h lies between 



1 + _i_\ and— L^|l - -;-\: (17.) 



hence, inasmuch as 



it follows at once that the probable error of a single observation 

 depends on the mean error by the equation 



r = g ^ 2 . £3 = 0-674489 £2 (18.) 



if the numerical value of g \/ 2 is substituted. The certainty 

 of this determination is given by the limiting values of h. It is 

 an even chance that r lies between 



\/ m !>/ in 



instead of which, as absolute exactness is not contemplated, vre 

 may permit ourselves to make the limits of r = 



We neglect in this the higher powers than the first of the un- 

 certainty of the probable error, considering the uncertainty as 

 a small magnitude of the first order. 



There still remains a circumstance to be attended to. The mag- 

 nitude £2 and with it h also, ought properly to have been deter- 

 mined from the true errors of observation, whereas it has been 

 only deduced from the obtained minimum of the squares of the 

 errors. It is clear that this mode of deduction is necessarily 

 somewhat faulty, as every value of x which differs ever so little 

 from the arithmetical mean must give a greater eg and a lesser h. 

 In order to gain a clearer view of this, let the most probable 

 value of X, as far as it follows from m observations, = p, or 



P — ^^^y 

 m 



VOL. II. PART VII. 2 A 



