24 



THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



It is tolerably manifest that these sums decrease faster for the large declinations 

 than for the small ones ; and we shall probably see the law more clearly by referring 

 the correction to declination 0° than to the mean declination. For this purpose sub- 

 tract from each sum the correction for declination 0°, and we have 



Declination . . 

 Corrected sum 



It appears that the numbers here are not very remote from the ratio of the squares 

 of the sines of the declinations. In fact, if we take the formula —-3168 sin^ h, we 

 have 



For declination 

 Formula .... 



3= 

 -9 



6= 

 -34 



27° 

 -654 



which may pass for a first approximation to the observed result, when it is considered 

 how much the errors are increased by addition. Dividing by 24, this gives — 132"™ sin^ S 

 for the correction to be applied to each result calculated for declination 0. When S 

 is about 16°, the mean value, this correction is 11"*. Hence U'" — 132™ sin'* § is the 

 correction to be applied to the mean value. This gives us the following Table, which 

 is a first approximation to Table XIX. of Mr. Lubbock. 



For declination 

 Correction ... 

 The sums 



6° 

 + 9 

 + 216 



Which agree nearly with 



+ 282 +250 +221 +205 +128 +30 



-90 



27° 

 - 16 

 ■384 



■151 —272 —382 



132 sin^S is when sin* 5 = 73 or S = 



12 



the observed results, with sufficient accuracy. 



We may observe that the expression 11- 



16° 45'. This is the mean value of A, because the correction is applied to the mean. 

 Therefore the expression is 132 (sin° A — sin^^). 



In each vertical column of Table XIX., the value appears to be greatest and least 

 when the time of the moon's transit is about 7^ and 1'^. Hence we shall take the 

 correction given by the above formula, and try whether the residual phenomenon, after 

 this correction has been applied, is governed by any fixed rule. 



For this purpose apply the above correction with an opposite sign to Mr. Lubbock's 

 Table XIX, The numbers in the columns are minutes. 



