324 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



and so on, according as we take w'=20, 40, 60, or still greater values. 

 It is seen that the first value, in which we get the value of P n , from 

 summing the actual values of K u from n=6 to n—n' f and then get the 

 sum of the remaining infinite number of terms approximately from the 

 last of (20), differs but little from the last value, in which the value of 

 P n , was obtained from summing the actual values of K n up to w'=60, 

 and then obtaining the sum of the remaining terms from the last of 

 (20). It is evident that the real value of u must be only a very little 

 less negatively than — 2.6870. The several values of u differ the less, 

 the more nearly the condition of (16) is satisfied, which, when the value 

 of n' is large, is very nearly that of (18). In our example /?=10, and 

 so is too large to give equal values in the several cases of w'=20, 40, or 

 60. With /?=40 there is much greater difference in the several values, 

 and the uncertainty in the last value is consequently much greater, 

 but the last number so obtained is always a limit below which the real 

 value is. 



Since our values of K n have been computed in terms of E the value 

 of u above must be multiplied into E. With this value, then, we get 

 from (6) for the value of h at the equator, where v^l, 



h= (1-2.687) #=-1.687 E. 



The value of E is that of the amplitude of the equilibrium tide at 

 the equator, which in the case of the lunar tide, if we assume the moon's 

 mass equal -^, is 0.812 of a foot. Hence we get for the range of the 

 lunar tide, approximately, at the equator, 



2 h=-2 x 1.687 x 0.812= -2.74 feet. 



Its being negative indicates that low water occurs at the time of the 

 moon's meridian transit. 



Laplace, in the same case, obtained for the range of the tide for the 

 mt-cn and sun in conjunction or opposition 11.05 metres, wffich, being 

 positive, indicates that high water occurs at the time of meridian pas^ 

 sage. Bin instead of iT 4 =0, he used .2^=6.190, obtained from his con- 

 tinued fraction. Besides, the mass of the moou which he used was 

 much too large. 



