264 



in which yl = M2, a = nio ; Bi, B2, etc., are the amphtudes of the other 

 harmonic components (except Mi and the lunar overtides) and 61, 

 62, etc., are the respective speeds of these components. 



Since the speeds of the kmar overtides are two, three, and four times 

 the speed of M2, the successive increments of x in equation (12A) 

 for these components, as n increases by successive integers, are 47r, 6^, 

 and Stt respectively. The successive values of the trigonometric 

 functions of x in that equation are therefore all identical. Similarly 

 the successive increments of x for the Mi component are each equal t, 

 and for the M3 component 3/2. ir For all of these components the 

 mean value of sin^ x is not }i The effect of these components on the 

 elevation of mean high water does not therefore follow the law 

 expressed by equation (14A). These components are however gen- 

 erally too small to affect the elevation of mean high water appreciably, 

 and the terms to be added to account for them need not be developed 

 here. 



8. Mean tidal range. — The elevation of mean low water below mean 

 sea level may be derived in the same manner as the elevation of mean 

 high water above sea level, and with the identical result. The 

 expression for the mean tidal range is therefore: 



Mn=2^[l + K(5r&i'MV+-52-62-A4-V+ • • •)] (15A) 



The factors 6l-/«^ ^2^/0^", etc., are close to unity for the semidiurnal 

 components, and close to ji for the diurnal. The ratios Bi^/A^, B2^/A'^, 

 are very small for those components whose amplitude is less than one- 

 twentieth of that of the M2 component. Omitting the components 

 that rarely if ever exceed tliis ratio, ecjuation (loA) becomes: 



Mn=2M2[l + }US'2s'2/M2'm22+N2W/M2'mo^^+K22k27M22m2' 



+ Ki^^i VM,2m.2 + Oi^oi VM.^m .' + Pi'p iVMo'm.,^ + Q,%' /M-Mi,') ] 



(16A) 



9. The numerical value of the mean tidal range derived from 

 equation (16A) is always substantially less than that derived from 

 direct observation. Aside from the effect of overtides and the approx- 

 imations introduced in the derivation of the formula, this deficiency 

 may be attributed to the fact that any accidental variation in the 

 water elevation occurring near the time of computed high water 

 increases the observed high water by substantially the maximum 

 amount of the variation if positive, but decreases the observed high 

 water by but substantially the minimum amount of the variation if 

 negative. In the long run, therefore, these variations effect a cumu- 



