338 H. Jacoby — Determination of the Division 



Division error of line m = 2K-mK 

 And for the other scale, 



m 



Division error of line m = — (2Q-mK ) 



3. The above is Gill's method, substantially in the form 

 described in the Monthly Notices (vol. xlix, p. 110) ; though 

 the way in which it is presented has been somewhat modified. 

 The formulae (1) are the same as Lorentzen's final formulse, 

 given in Astr. Nach., 3336. 



The next step is to calculate the weights of the division 

 errors determined. This can be done very easily, if we notice 

 in Table I that the sum of any number of consecutive paren- 

 theses in the same horizontal line is known with the same pre- 

 cision as any one parenthesis alone. Thus the quantity : 



(«.-*,) + (<*-<>.) + (*,--&.) 



is known with the same precision as the equivalent quantity : 



This circumstance will of course affect the sums of the K's, 

 the mean error of 



for instance, being by no means as great as 



V2 (mean error of K x ), 

 as it would be if K, and K 2 were independent. Now let us 

 put: 



s = mean error of any of the observed quantities, as c v # 2 , etc., 



which are supposed to be all observed with equal precision. 

 A m = division error of line m on scale A. 

 s m = b m + c m + d m + . . . , 



so that s m is the sum of all the letters except the first and last, 

 after each has received the subscript m. Then we get at once 

 from equations (1) 



n—m ™ m « 

 A m = ^ K 2 K 



n 1 n m +i 



Also we see from Table I that 



ni'K = 5 -5 ro +(e + e 1 + . . . +^_i)-(a 1 +a s + • '. • +<*») 



1 



n2K = s m —s n + (e m + e m+1 + . . . +e n _ 1 )-(a m+l + a»+ 8 + ... + «») 

 Substituting these values in the previous equation gives : 



