AT KEW OBSERVATORY AND THEIR DISCUSSION. 



451 



Formula for the Variation of the Differences of the Descending and Ascending 



Headings throughout the Range. 



9. The absolute size of the differences between the descending and ascending 

 readings varies largely from aneroid to aneroid ; thus formulae reproducing the data 

 of Table I. would not be immediately useful. On the other hand, the law of 

 variation, as shown by the ratios of these differences, at different points of the range, 

 to the mean difference, appears to be nearly the same in all ordinary aneroids. I 

 have thus tried to represent the variations shown in Tables II. and III. by simple 

 algebraic formulae of the type 



(1), 



y 



3 x 2 -f- 



where y is the ratio of the difference of the descending and ascending readings to 

 the mean difference, x the fraction of the range measured from the lowest point. 



One can secure absolute agreement between the observed and calculated values in 

 such a case by taking enough terms ; but, for practical purposes, the question is 

 whether with a comparatively small number of terms one can secure a sufficiently 

 close agreement. In the present instance, four terms are enough for practical 

 purposes. To get the best general agreement we ought, of course, to determine , 

 i, 2 , a 3 by the method of least squares, taking account of the whole eleven values 

 in each range. In dealing, however, with material such as that here available, the 

 slightly increased accuracy thus attainable would not, in my opinion, be an adequate 

 return for the labour expended. I thus simply determined the constants so that the 

 observed and calculated values were identical for the values '1, '3, '7, and '9 of x. 

 The values thus found for the constants are given in Table IV., the first six sets of 

 values relating to the old observations of Table II., the last set to the mean data of 

 Table III. 



TABLE IV. Values of Constants in y = + a \ x + ff-a* 2 + 



The value of a, shows little variation. In the shortest range the values of 2 

 a 3 are undoubtedly unduly large numerically, and in the range 30-1 8 inches there is 



3 M 2 



