Errors of a Straight Scale. 341 



n is an uneven number, it applies to either of the two lines 

 nearest the middle of the scale. 



4. Having now described Gill's method, I shall proceed to 

 show how it may be improved very materially. For this pur- 

 pose, let us resume the equation : 



A m =— a — - s 4- __e — _-« +-U n -2e+2fl — 

 n L o i J n L m +1 J 



Now let us suppose that the various observed quantities con- 

 tained in Table I, have been determined with different weights 

 as follows : 



Weight of a = ^- 

 Weight of c 2 = — 



C 2 



etc. etc. 



In other words, let us indicate the number of times each letter 

 is observed by the reciprocal of that letter primed. Then if 

 we put : 



S' m =?>'m + C' m + Cl f m + . . . 



following the analogy of s m , already employed, we shall have: 



(mean error of s ) 2 = s' ££ 

 (mean error of s m y = s' m S£ 



(mean error of _5 e)*= es 2 e f 







etc. etc. 



Consequently we shall have for computing the square of the 

 mean error of line m on scale A, assuming es = 1 : 



n\EE) m = n'— = »V. + (»-m)T V.+'j? e' '+ _§ a' 1 



±m L o 1 J 



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



L m «H-l J 



This equation may be used in a very simple way for strength- 

 ening Gill's method. Let us put the weights of all the exterior 

 quantities in Table I equal to each other, i. e., let us put : 



2_i i _ i __ _i._. i __ i __ i __l_i _ i __i __.iT-.-JL 



b'~c'~ d'~ e'~~ e'~ e'~ e'~ d' ~ c' ~V ~ a'~ a' ~ a' n ~~ a' 



and suppose any one of these quantities to be represented by 

 — . In other words, let us observe every comparison between 

 the two scales in which either of the four end lines appears — 



