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



but let the true value he p + A p. By substituting j9 in the 

 equations of condition, we obtain as the errors of the observations 

 the magnitudes p — n, p — n',p — n" ... ., which for the sake 

 of brevity may be designated by a., «', a". The substitution 

 of the true value p + A p would have given p + A p — n, 

 p -\- Ap — n' , p + A]J — n". . . ., and these latter magnitudes, 

 which may be called 8, S', 8", would have been the true errors of 

 observation. We have consequently the equations 



« + A p = i 



cc' + Ap = ^' 



«" + Ap = 8", &c. 



As [«] = Oj the sum of the squares taken on both sides will give 

 [a^] + m Ap'^= [8-]. 

 Thus if we assume [a®] as the true sum of the squares of the 

 errors, we shall always err by the positive magnitude m A p^. 

 This representation gives at once the means of correcting the 

 error as far as circumstances permit. If to the m observations a 

 new one were added without our knowing determinately what 

 value it had given, we should have to add to the [a-] the value 

 e/ as the mean value of such a square. The equation shows 

 that m A jt>^ must be added in every case ; and it follows from 

 what has been said above, that p has the weight m, or that if a 

 single observation has the mean error Sg 5 the mean error of p 



will be equal to —j- . Hence it follows, that we approach the 



truth as nearly as possible, if in this equation we take the mag- 

 nitude of A JO such as its proportion to the single observations 



gives it, or if we substitute the value Ap — -~ . Then 

 ° ^ s/ m 



[«^] +./=[8'^], ^ 

 and as it follows from the assumed definition that 



the value of Sg derived from the m errors remaining over after 

 the substitution of the arithmetical mean is obtained by 



(m- 1) . s/ = [«'J. (20.) 



In order to obtain as nearly as possible the true mean errors 

 of the observations, we must, with an unknown magnitude, regard 

 the sum of the squares of the errors as if it belonged not to m, but 

 to (m — 1) errors. 



