26 



THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



(sin' S — sin' A) (l32 + D sin 2 (^ - y)). 

 We have to find D, For that purpose take the formula 



(sin' h - sin' A) D sin 2 (^ — y), 



which expresses the residual phenomenon just given from Table XIX. Take the case 

 where § = 0, and we have for the first vertical column, the expression 



— D sin' A sin 2 {(p — y). 

 The vertical column contains all the values of this for every half-hour of the value 

 of (p — y, that is, for values of sin 2 (<p — y) taken at intervals of 15° round the cir- 

 cumference. Taking" the sum of these values for one semicircle, it is, by known 

 formulae, 



!!ilS2!.i<|llM! = tan 82F = 7-5957. 



Now this sum in the Table is 45 if we take the mean of the positive and negative 

 values ; observing, however, that this value compared with the succeeding columns 

 appears to be smaller than the general course of the numbers would give it. Hence, 



D sin' A X 7'5957 = 45 ; D = 72 nearly. 



Hence D sin' A = 6. But it will agree better with the general numbers to make D = 7, 

 and the expression for the residual phenomenon is 



84 (sin' h — sin' A) sin 2 ((p — y). 



The values of 84 (sin' A — sin' I) for the successive values of I are hence found ; and 

 hence the corrections. 



Assuming y = 4^, the following Table represents the table of the residual pheno- 

 menon. 



Table of the Expression 84 (sin' A — sin'S) sin 2 ((p — y). 



This agrees as to its changes of magnitude and sign, and as to the mean of the 

 numbers, with the table of the residual quantities, p. 25. The formula, 



(sin' I — sin' A) { 132 + 84 sin 2 (^ — 4^)} 



