22 



THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



On examining this Table, it appears that in the column corresponding to H. P. 5/', 

 the differences from the mean, or corrections for parallax, are very small for all hours 

 of the moon's transit, (ranging from + 5°^ to — 4",) and that the positive nearly ba- 

 lance the negative values. We may suppose, therefore, that for H. P. 57' nearly, the 

 correction is 0. It appears also that the correction is generally negative when the 

 H. P. is greater than 57', and positive when it is less, the exceptions being of small 

 amount compared with the general mass of observations ; and if we take the sums in 

 each vertical column of Table XVII. we shall find that they are nearly as the differ- 

 ence of the parallax from the mean value 57'. It appears, therefore, that this correction 

 must involve a factor (P — p), when P is the mean horizontal parallax of the moon 

 (or 570} and JO any other value of her horizontal parallax. 



If we take any vertical column of this Table, and thus follow the correction through 

 the various hours of the moon's transit, we find that for all values of the parallax the 

 correction is very small, when the moon passes at 5'' 30™ or 6^, and that the positive 

 and negative values in that case nearly balance each other. In each column, when 

 the hour of transit is either greater or less than this, the correction increases with the 

 difference of hour, and proceeds to a maximum, which appears to occur about 9** or 

 10^ transit. As a simple way of satisfying these conditions, we may suppose the cor- 

 rection to involve the factor sin ^((p — j3) when |3 is a constant quantity : and com- 

 bining this factor with the one already found, we shall have B (P — js) sin ^{<p — |3) 

 for this correction in minutes of time. 



It appears that in order to give the maximum value of this correction when it 

 occurs at about 10^, |3 must not be much different from 4'\ In order to determine B, 

 take the formula B (P — jo) sin^ ( (p — |3) for every half-hour : its value is 



2 B (P - ;?) {sin2 7J° + sinS 15° + sinS 22i° + sin^ 30° + sin^ 374° + sin^ 45° 

 + sin2 82 J° + sin2 75° + sin^ 67 J° + sin? 60° + sin^ 524°} 



= llB(P-jo). 



Comparing this with the sums for H. P. 54', 55', 56', 57', 58', 59', (the other columns 

 being incomplete,) we have. 



Horizontal parallax 



Formula 



Observed sums . . , 



59' 

 - 22B 

 -112 



Hence, taking the sums, 99 B = 595, whence B = 6, and the expression is 6 (P — p) 

 sin2(^~4»'). 



This may be put in the form 3 ( P — j9 ) ( 1 — cos 2 (^ — 4*^) ), 



or 3 (P -/?) (l + sin 2 ( ^ - l^") ) . 



The agreement with the sums observed is as follows : 



Horizontal parallax , 



Formula 



Observed sums, . . . 



59' 

 ■132 

 112 



