AT KEW OBSERVATORY AND THEIR DISCUSSION. 455 



The value assumed in the above illustration is that found in Table I. for the range 



30-15 inches. Hence the values just deduced for <F 4 , &c., are what the mean 

 differences for the other ranges would have been in the case of the average aneroid 

 sent for trial over the range 30-15 inches. Comparing these calculated values with 

 the mean differences actually recorded in Table I., we should draw the conclusion 

 that the average aneroid sent for testing over the longest range is less subject to 

 after-effect than the average aneroid sent for testing over any other range except the 

 shortest. This is, I believe, really the case. The exception in favour of the shortest 

 range is due to the exceptional size of a number of the aneroids intended to cover the 

 range 30-26 inches. 



13. The sum of the differences of the descending and ascending readings is 

 really the source of our knowledge of the mean differences, and for some purposes it 

 may well be the more convenient quantity of the two. The formula for it is 

 deducible at once from (5), for 



Sum of differences when range n inches __ , + !,, , 

 Sum of differences when range 9 inches 10 



= -0028 + -0025n + -0060rr + -00068n 3 . . .(6), 



= \ji (n), say. 

 The following results are easily verified : 



= -147, ^(6) = '375, t/(9)=l, ^(12) = 2-066, ^(15)=3'680. 



Using S, t to denote the sum of the differences for a range of n inches, then, according 

 to the formula, 



S n -5-tJr(n) = 8 w -5-iJr(m) 

 for all values of n and m. 



To illustrate the degree of accuracy attained by the use of such formulae, I have 

 taken the case of the first 24 special experiments, and determined the arbitrary 

 constant for each aneroid so as to make the total sum of the differences over all the 

 ranges for which the instrument was tried the same for the calculated as the 

 observed values. As it may be well to show how this was done, take the case of 

 aneroid No. 2. Here the sums of the differences observed in the three shortest 

 ranges were respectively '163, '443 and 1'228 inches, amounting in all to 1'834 ; 

 while by the formula, 



*(9)= 1-522. 



Thus, for a range of n inches with this aneroid, we may assume 

 Sum of differences = </ (n) X 1'834 -=- 1'522. 



