HAEMONIC ANALYSIS AND PREDICTION OF TIDES. 119 



predictions are made by applying to the times of the moon's transits 

 and to the mean height of the tide systems of differences to take 

 account of average conditions and various inequalities due to changes 

 in the phase of the moon and in the declination and parallax of the 

 moon and sun. 



Without the use of a predicting machine the harmonic method 

 would involve too much labor to be of practical service, but with 

 such a machine the harmonic method has many advantages over the 

 nonharmonic systems and is now used exclusively by the U. S. Coast 

 and Geodetic Survey in making predictions for the standard ports of 

 this country. 



The height of the tide at any time may be represented harmonically 

 by the formula 



A = i7o + 2/S'cos [a«+(Fo + w)-/c] (465) 



in which 



^ = height of tide at any time t. 

 fi^o = mean height of water level above datum used for pre- 

 diction. 

 i7=mean amplitude of any component A. 

 /= factor for reducing mean amplitude H to year of pre- 

 diction, 

 a = speed of component A. 



t = time reckoned from some initial epoch such as beginning 

 of year of predictions. 

 ( Vo -f u) = value of equilibrium argument of component A 

 when t = 0. 

 K = epoch of component A. 



In the above formula all quantities except h and t may be con- 

 sidered as constants for any particular year and place, and when these 

 constants are known the value of li, or the predicted height of the 

 tide, may be computed for any value of t, or time. By comparing 

 successive values of h the heights of the high and low waters, together 

 with the times of their occurrence, may be approximately determined. 

 The harmonic method of predicting tides, therefore, consists essen- 

 tially of the application of the above formula. 



The exact value of t for the times of high and low waters will be 

 roots of the first derivative of formula (465), equated to zero, which 

 may be written — 



j^ =- -EafH sin [at -{-{Vo + u)-k] = (466) 



Although formula (466) can not, in general, be solved by rigorous 

 methods, it may be mechanically solved by a tide-predicting machine 

 of the type used in the office of the U. S. Coast and Geodetic Survey. 



The constant He, of formula (465) is the depression of the adopted 

 datum below the mean level of the water at the place of prediction. 

 For places on the open coast the mean water level is identical with 

 mean sea level, but in the upper portions of tidal rivers that have an 

 appreciable slope the mean water level may be somewhat higher than 

 the mean sea level. The datum for the predictions may be more or 

 less arbitrarily chosen, but it is customary to use the low-water plane 

 that has been adopted as the reference for the soundings on the 

 hydrographic charts of the locality. For all places on the Atlantic 



