Errors of a Straight Scale. 337 



K,= i[(e„-<*,) + «-c,) + («,-*,) + (6 .-«,)] 



It is evident that K 1 is the excess of the space 0, 1 over the 

 true fourth part of scale B, since it is the mean of the 

 excesses of 0, 1 over the four constituent spaces of scale B. 

 Consequently, if we assume scale B as standard, the space 0, 1 

 is too large by the quantity K x , which we may express as fol- 

 lows : 



Division error of line 1, scale A 1 = K x 



Similarly the space 1, 2 is too large by K 2 , and therefore the 

 space 0, 2 is too large by the quantity K x +K 2 ; or: 



Division error of line 2, scale A, = Kj4-K 2 

 Similarly : 



Division error of line 3, scale A, = K x +K 2 + K 3 

 Division error of line 4, scale A, = K 1 + K 2 + K 3 + K 4 



Since we have assumed scale B as the standard, the quantity : 



K.+K. + K. + K. 



or the division error of line 4 of scale A, is really the differ- 

 ence in length of the two scales ; so if we put : 



K = i(K 1+ K, + K 8 + K 4 ), 



K will be the excess of the true average space of scale A over 

 the true average space of scale B. Consequently, if we wish 

 the division errors of scale A to be expressed in terms of that 

 scale itself as a standard, we mast write : 



Division error of line 1 — K l — K 

 Division error of line 2 = Kj + K,, — 2K 

 Division error of line 3 = K^ + Kg + Kg — 3K 

 Division error of line 4 = K x + K 2 + K 3 + K 4 — 4K = 



Table I also furnishes the corresponding division errors of 

 scale B. If we take diagonal means as follows : 



Q« = ![(«.-*,) + (« -<« +(«,-<*,) + («.-*.)! 

 0. = i[K-e.) + C*,-«J + (*-.«.) + «-«J] 



etc. etc. 



we have : 



Division error of line d= — (Q d — K ) 

 Division error of line c = — (Q d + Q e — 2K ) 

 Division error of line b = — (Qd + Q c -t-Qj — 3K ) 

 Division error of line a = — (Q f 7 + Q c + Qi + Qa— 4K ) = 



since, necessarily, 



K 1 + K 2 + K 3 + K 4 =: _(Q, + Q c + Q i + Q a ) 



Generalizing these formulae we have for the division error of 

 the line numbered m on scale A, supposed to have n+1 lines, 

 numbered from to n, the following : 



