342 H. Jacoby — Determination of the Division 



times instead of once, and use the mean in the further compu- 

 tations. We then have, since se = 1 : 



s' m = n—1, s' = (n— %, s' n = (n—l)q 



171—1 m n — 1 n 



2 e' = mq, ^ a' = mq, 2 e' = (n—m)q, ^ a! — (n—m)q, 



1 m m+1 



So that if we put all the other weights equal to unity, equation 

 (5) becomes : 



11 pi 2m ~~i 



_ = _ (n _ 1)+ L_ (w _ 1)+ _ ( „j 2 (6) 



This equation agrees with (2) if we put unity for q. 



I shall call Gill's method, when modified in the manner just 

 explained, method 1. In Table II given below, will be found 

 the maximum and minimum weights belonging to method 1, 

 placed for comparison next to those belonging to Gill's method. 

 In order to bring out more clearly the advantage of method 1, 

 I have also put in the table the total number of observations 

 required for carrying out Gill's method, and the additional 

 number required for method 1. This number is : 



For Gill's method, (n+ l) 2 — 2 

 Additional for method 1, (4^ — 2) / 1 J 



In computing the column " Time used " I have taken as the 

 unit the time required to make one observation in Gill's 

 method. I have then assumed that the additional observations 

 of method 1 each require only one-fourth as much time, since 

 they simply consist in additional settings after the microscope, 

 etc., have been brought into position and adjusted. In com- 

 puting the table, I have put : 



1 



i =4 ' 



which is equivalent to observing all the comparisons involving 

 the end lines four times. It does not seem desirable to carry 

 the number of observations beyond four for any one compari- 

 son, since it is doubtful if the mean of a large number of 

 observations of this kind really possesses a weight correspond- 

 ing to that number. 



For the sake of comparison, I set down here the theoretical 

 weights which would result from a rigid least square discussion 

 of the observations in Gill's method, according to Lorentzen. 



