344 H. Jacoby — Determination of the Division 



1 I m — l m ~~| 



n*=- = fl ' 2 s' m +(n-my \ s' + 2 e' + 2 a' 



+ m' i [s' n + n ^e' + 2a'~\ (5) 



and put m-f-1 for m in it. This gives : 



n 4 =p— =^V m+1 +(yi-m-l) 2 5' + ^ +^a' 



■^CT+l L 1 _J 



«— i 



+ (m + l) 2 ["*'„ + ".2e' + JSa'l 



v m+l m+2 _J 



In order to bring about equality of weights, we have only to 

 make : 



np-=n :p— 



r m -t m+1 



If we do this, and let — have the same signification as before, 

 we get, after some simplification : 



s' m -s' m+l =%[n-l + 2m(n-2)] (1) 



This equation shows that the sum of the reciprocals of all the 

 weights (except the top and bottom weights) in any column of 

 Table I exceeds that in the following column by the quantity : 



2 ^[7i-l + 2m(n-2)] 



And if this equation be satisfied for all values of m from m=l 

 to m=n— 2, the division errors will be determined with equal 

 weight. If we substitute successively 1, 2, 3, . , n—2 for m in 

 equation (7), we get a series of quantities in arithmetical pro- 

 gression. Summing this we derive : 



s'-s' m =- 2 ^(n-l) + ^[>(n-2) + l]m (8) 



lb It 



This equation shows that s\ can never exceed s' m by a quantity 

 greater than 2qn ; and since an inspection of Table I shows 

 that s\—s' m cannot possibly be made greater than n— 1, it fol- 

 lows that 2q cannot be taken greater than unity, or q greater 

 than J. We have already explained that it is not desirable to 

 make q less than \. It will therefore be best to put as before : 



It is obviously desirable to make : 



S\ = 71 — 1 



or in other words, to observe once all the quantities in Table I 

 that have the suffix 1, except the top and bottom ones e x 



