158 







Between No. 1 and No. 4 we have an interval of 37.85 years, and 

 a mean annual secular change of 2'-69 ; mean epoch, 1840.6. 



Between No. 1 and No. 2, comprising an interval of 16.65 years, 

 we have a mean secular change of 2'* 77 ; mean epoch, 1830.0. 



Between No. 2 and No. 4, comprising an interval of 21.2 years, we 

 have a mean secular change of 2'-63 ; mean epoch, 1848.9. 



Hence we may infer that the yearly diminution of the Dip from 

 secular change, though very nearly uniform throughout the whole 

 interval of 37.85 years, was somewhat greater in the earlier part of 

 the interval than in the later ; and that the rate of diminution 

 may admit of being more exactly represented hy the introduction of 

 a second term. 



If then we take the year 1840.0 as a convenient middle epoch =t , 

 and call its dip ; and if we further call the observed dip at the 

 several observational epochs t 19 t 2 , t 3 and 4 , respectively 6 V 2 , 3 , 4 , 

 we shall have four equations of the form 



and giving double weight to the equation furnished by the epoch 

 1859.5, inasmuch as it is derived from so much greater a body of 

 observations than the results at the other three epochs, we obtain by 

 least squares, 



=69 ll'-95; #=-2'713; y=+0 r '00057. 

 Hence we have the general formula for computing the dip between 

 the years 1820 and 1860, 



0=69 ll'-95-2'-713 (*-# )+0'-00057 0-*o) 2 > 

 t Q being 1840.0, and t being any other time for which the dip is 

 required. 



Using this formula, we have the differences between the com- 

 puted and the observed dips at the several epochs of observation 

 as follows : 



Computed. Observed. Computed Observed. 



1821.65 ...... 70 03-6 70 03-4 +0-2 



1838.3 ...... 69 16-8 69 17'3 -0-5 



1854.65 ...... 6833-4 6831-6 +1-8 



1859.5 ...... 6821-2 6821-5 -0-3 



And the dips corresponding to every tenth year within the period 

 specified are as follows : 



