20 THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 



in which X' is the mean interval of the tide and transit, and d' the correct interval ; <p 

 the solar time of the moon's transit, and u a constant quantity. The ratio of the 

 quantity h' to h is 2-9884 : 1 ; the quantity a is 2 hours. 



According either to the method of Bernoulli or to that of Laplace, there would 

 result from the theory an expression of the above form, for the interval of tide and 



moon's transit. By assuming suitably the values of -^ and a, the results of obser- 

 vation at other places may also be made to agree very closely with the above formula. 

 The curves which represent by their ordinates the successive values of the above for- 

 mula, when constructed for different places, exhibit a remarkable general similarity, as 

 may be seen in the Philosophical Transactions, 1831, where Mr. Lubbock has given 

 these curves for Portsmouth, Plymouth, Sheerness, London and Brest. The curve is 

 symmetrical with respect to the axis, intersecting it when <p = 05, and when (p = a + 

 a fourth of a circumference. Its ordinate has a negative minimum and a positive maxi- 

 mum, which are equal in magnitude ; but these values are not midway between the 

 values 0, consequently the ordinate increases more rapidly after the minimum and 

 before the maximum, than it diminishes before the minimum and after the maximum. 

 This property appears very clearly in the curves constructed for all the above ports. 



But in other respects the result of the observations, thus compared, does not agree 

 with the theory. According to the theory, the quantities h and h' express the amount 

 of the separate solar and lunar tides respectively, and as the ratio of these effects must 

 be the same for all places, the maximum value of the semimenstrual inequality ought 

 to be the same in all the above cases ; namely, the time corresponding to half the 



h h' 



angle whose tangent is — : . If, as Laplace finds from the Brest observations, j- 



= 2'6157j the angle corresponding to the above tangent is 22° 28'; the maximum 



value of the inequality is 45", and the double of this, or 1^ 30™, is the difference of 



the greatest and least interval of the tide and moon's transit. 



According to observation, the difference of the greatest and least intervals is as 



follows * : 



London 1^ 28™ 



Sheerness 1 29 



Portsmouth 121 



Plymouth 1 36 



Brest 1 19 



It appears unlikely that the difference in these values for Plymouth and Brest, or 

 even Plymouth and Portsmouth, can depend upon accidental causes, or too limited a 

 number of observations. It would appear, therefore, that the coefficient of the semi- 

 menstrual inequality, \jr\ is different at different places ; a circumstance which no 

 extant theory would have led us to expect. This subject, however, deserves further 

 * We suppose here the eifects of parallax and declination to be eliminated by the averages of the observations. 



