THE LA W OF ERROR. 449 



determined, this method evidently resolves itself into 

 taking the mean of all the values given by observation.' 

 Encke, again, distinctly says m , that the expression for the 

 probability of an error 'not only contains in itself the 

 principle of the arithmetical mean, but depends so imme- 

 diately upon it, that for all those magnitudes for which 

 the arithmetical mean holds good in the simple cases in 

 which it is principally applied, no other law of proba- 

 bility can be assumed than that which is expressed by 

 this formula/ 



It can be shown, too, in a moment that the mean is the 

 result which gives the least sum of squares of errors. 

 For if a, b, c, &c., be the results of observation and x the 

 selected mean result, the sum of squares of the errors is 

 (a x) 2 + (b x) 2 + (c #) 2 + &c., which is at a minimum 

 when its differential coefficient 2 (a x + b x + c x + 

 &c.) = o. From this equation we immediately obtain, de- 

 noting by n the number of separate results, a, b, c, &c., 

 x = (a + b + c + . . . ) -, or the ordinary arithmetic mean. 



Weighted Observations. 



It is to be distinctly understood that when we take the 

 mean of certain numerical results as the most probable 

 number aimed at, we regard all the different results as 

 equally good and probable in themselves. The theory 

 of error expresses no preference for any one number over 

 any other. If, then, an observer has reason to suppose 

 that some results are not so trustworthy as others, he 

 must take account of this difference in drawing the mean. 

 By the method of weighting observations this difference of 

 value is easily allowed for. Astronomers are in the habit 



m Taylor's ' Scientific Memoirs,' vol. 5i. p. 333. 



