324 THE MECHANICS OF THE EARTH's ATiMOSPHERE. 



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

 It is seen that the first value, iu which we get the value of P„. from 

 summing the actual values of K„ from m=G to n = n'y 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, iu which the value of 

 P,^, was obtained from summing the actual values of K,^ up to 7i'=00, 

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

 (20). It is evident that the real value oX' 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 /i=10, and 

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

 60. With y5=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 r<'al 

 value is. 



Since our values of K,^ have been computed iu terms of E the va ..^ 

 of u above must be multiplied into U. With this value, then, we got 

 from (0) for the value of h at the equator, where y=l, 



k= (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 g^, is 0.812 of a foot. Hence we get for the range of the 

 lunar tide, approximately, at the equator, 



2h=-2x 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 

 moon and sun iu conjunction or opposition 11.05 metres, which, being 

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

 sage. But instead of Ki=0, he used Jr4=6.196, obtained from his con- 

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

 much too large. 



