SECT. 16.] Theory of the Average. 481 



the same thing for our present purpose, was the ultimate 

 average of all the measurements made. What we mean by 



a symmetrical arrangement of the values in regard to 0, is 

 that for every error OB, there shall be in the long run a pre 

 cisely corresponding opposite one OB ; so that when we erect 

 the ordinate BQ, indicating the frequency with which B is 

 yielded, we must erect an equal one B Q . Accordingly the 

 two halves of the curve on each side of P, viz. PQ and PQ 

 are precisely alike. 



It then readily follows that the secondary curve, viz. 

 that marking the law of frequency of the averages of two or 

 more simple errors, will also be symmetrical. Consider any 

 three points B, C, D: to these correspond another three 

 B , C , D . It is obvious therefore that any regular and sym 

 metrical mode of dealing with all the groups, of which BCD 

 is a sample, will result in symmetrical arrangement about 

 the centre 0. The ordinary familiar arithmetical average is 

 but one out of many such modes. One way of describing it is 

 by saying that the average of B, (7, D, is assigned by choosing 

 a point such that the sum of the squares of its distances from 

 v. 31 



