ON THE COASTS OF IRELAND. 23 



high water does not differ much from the sun's hour-angle at the time of solar diurnal 

 high water. The assumption of any constant difference, either =0 or having any 

 assigned magnitude, reduces two of the unknown quantities to one. 



The number of unknown quantities is thus made the same as the number of data, 

 and the solution can therefore (speaking in a strictly algebraical sense) be effected, 

 in general. 



The following is the method by which the equations for the four unknown quanti- 

 ties may most conveniently be formed : — 



From the ordinary facts of the tides, it seems probable that the coefficient of lunar 

 diurnal tide may depend on the moon's declination at a few days, perhaps not 

 exceeding five, anterior to the time of the tide. Let d^ be the moon's declination at 

 one day preceding the time of tide, c^^ the moon's declination at three days preceding 

 the time of tide. Then we may express the coefficient of lunar diurnal tide by 

 />.sinrfi4-y.sin6/3; where by varying the proportions of/? and q the coefficient may 

 be made to depend on the moon's declination at any day near them ; and by varying 

 the magnitudes oi p and q in the same proportion, the magnitude of the coefficient 

 will be altered in that proportion. 



The coefficient of solar diurnal tide may be represented with sufficient accuracy by 

 r.sinDi, where Dj is the sun's declination one day preceding the time of tide. 



Let h be the solar hour of the tide. This is the same as the hour-angle of the sun 

 to the west of the meridian. The phase of the solar diurnal tide will depend upon this 

 angle diminished by some unknown constant s ; and the elevation of the solar diurnal 

 tide may be represented by S.sinD^.cos A— 6-. 



Let t be the moon's time of transit. Then the moon's hour-angle west of the 

 meridian is h—t. Therefore if the phase of lunar diurnal tide depended on the 

 moon's hour-angle in the same manner in which the phase of solar diurnal tide de- 

 pended on the sun's hour-angle, the elevation of the lunar diurnal tide would be 

 represented by {p.%\ndy-\-q.&md^co&h—t—s. But we know by the retardation of 

 the period of spring tides, as well as by the theory of tidal waves affected by friction*, 

 that in semidiurnal tides the lunar wave is more advanced in its phase with regard 

 to the moon's hour-angle than the solar wave is with regard to the sun's hour-angle. 

 We may conjecture, by analogy, that the same holds for diurnal tide. Putting a for 

 this difference of advance of phase, the elevation of the lunar diurnal tide will be 

 represented by (/>.sin d^-^-q.^in d.^)cosh—t — s-\-K. And the compound effect of lunar 

 and solar diurnal tides, expanding the cosines, will be 

 S.sin Di.(cos h.cos .s-f-sin ^.sin s) 



-\-(p.sm di-\-q.s\n d^)(cos h—t-{-u.cos s-{-sm h—t-\-u.sm s). 



Let S.cos s=w, S.sin A=^, ^=i/,-^=z ; and the expression becomes 



* Encyclopaedia Metropolitana, Tides and Waves^ Art. 326. 



