Errors of a Straight Scale. 345 



and a v By means of equation (8) we can then find out how 

 many times to observe the quantities having the suffix m. Of 

 course we cannot apply equation (8) rigorously, as that would 

 make it necessary to observe some of the quantities a fractional 

 number of times. In fact, if we observe even one of the 

 quantities in column m twice instead of once, we shall have 

 made: 



and we cannot possibly make it less than J, unless we leave it 

 zero. It will therefore be best to begin to make 



s'-s' m '-=i 



as soon as its theoretical value exceeds J, and to continue 

 it -J, until its theoretical value reaches f . We can then make 

 it 1, until its theoretical value exceeds J-, etc. To do this con- 

 veniently, we- must solve equation (8) for m, after putting 

 q = J. This gives, finally : 



m = Vl(n-l) (w-2) + 8n>-2) «-Q + 1 -1 ( 9) 

 2(n—2) 



We shall now apply this to an example. Suppose n = 4, as 

 in Table I. Equation (9) then becomes : 



_ V25 + 25Q(s' 1 -s' m ) -1 

 m- 4 



from which we get : 



for s\—s' m = \ m = 2, I 



" <-*'. = f m = 3, 4 



We should, therefore, in Table I begin to observe one of the 

 interior quantities twice for m = 3. For this purpose we 

 might select either b 3 or c r 



The above method 2 renders all the weights almost exactly 

 equal. What the common weight is can be computed from 

 equation (5). Since the weight is always the same, we may 

 put in equation (5) : 



m = 1, s' m = n — 1 

 We, therefore, also have : 



s 'o = {n-\)q s' n = (n—l)q 



2e'= q J2e'= {n—\)q 



m 



2a f = q 2 a'= (n — \)q 



1 in— 1 



Consequently : 



1 -*- 1 . r >-i)K-i) 3(n-in 



Am. Jour. Sci. — Fourth Series, Vol. I, No. 5. — May, 1896. 

 23 



