Errors of a Straight Scale. 339 



Am = »T«r., + !iS_ 2a 1 + "1%, - 3W i a 1 -> 



n 2 L o i J ■ n L » »+i J w 



This equation is not intended for calculating A m , since equa- 

 tion (1) is more convenient for that purpose. But in the 

 present equation, all the quantities are independent, so that it 

 can be used at once for getting E, the mean error of A m . Since 

 e is the mean error of any one of the observed quantities, we 

 shall have : 



(mean error of s ) 2 = (n—l)es 



TO— 1 



(mean error of 2e) 2 =i?iss 





 n— 1 



(mean error of 2e) 2 =: (n—m)ee 



(mean error of s m y = (n — l)ee 

 etc. etc. 



Consequently, 



EE= \ 1^^ I |rc — 1 + m + m | > se 



+ | — n — 1+ n — m + n—m H — I £6 



= 2 tf{n — l)-f m(n — m) — 

 If, therefore, we assume our unit of weight such that : 



68 = 1 



we shall have for computing the weight P m of line m on scale 

 A, the equation : 



1 2(n — l) 2m, 



K= *r + * ln - m) (2) 



— has a maximum value for : 



n 

 ra = - 



We therefore have 



Onn , _ 



1 , , 2n—£ 



-p- (maximum) = — ^ (3) 



The minimum value of ^-, subject of course to the condition 



that m may only vary from 1 to ^— 1, will occur when : 



m = 1 or m = n— 1 



In either case : 



* -/ ". . x 2(n-l)(n 2 +l) '' 



=p (minimum) = — '-\ ! — - (4) 



