TIDAL OBSERVATIONS. 297 



Hence if we take ^, 33, C, J3, (£ to denote five integrals depending solelj^ 

 on the distribution of land and water, expressed as follows : — 



%=—ff co&H cos 2\diT, '^= —ff cos^Z sin 2Xd(T, 



C=—ff8mlcoalcoa\dff, ^=—ffsmlcoBlBm\da, )► . . (21). 



(!£=l/f(3amH-l)dff,^ 



"where of course cZ(r=cos IdldX, 

 we have 



^ar{|co3'^(Heos2i//+3Ssin2;//) + 6sin2eosS(Ccos;// + iisin;//)+i(£(3sin2S-l)}(22). 



This, used with (19) and (20) in (17), gives for the full conclusion of the 

 equilibrium theory, 



i«r(3 8in2Z-l-®)(3sin2a-l) -^ 



+ 2ar[(sinZcoaZcos\— C) cos»/' + (8inZcosZsinA— J3)sin»//]sinScosS I „. 



r ' (•^'j)> 



+ ^[(cos'Zcos2\-a)cos2i/;+(cos' I sin 2\-a8) sin 2;//] cos' S 



in which the value of r may be taken from (18) for either the moon or the 

 sun ; and d and \p denote the declination and Greenwich hour-angle of one 

 body or the other, as the case may be. In this expression we may of course 

 reduce the semidiurnal terms to the form A cos(2;^ — e), and the diurnal 

 terms to A' cos (\p — e'). Interpreting it we have the following conclusions : — 



" 809. In the equilibrium theory, the whole deviation of level at any point 

 of the sea, due to sun and moon acting jointly, is expressed by the sum of 

 six terms, three for each body. 



" (1) The lunar or solar semidiurnal tide rises and falls n proportion to a 

 simple harmonic function of the hour-angle from the meridian of Greenwich, 

 having for period 180° of this angle (or in time, half the period of revolution 

 relatively to the earth), with amplitude varying in simple proportion to the 

 square of the cosine of the declination of the sun or moon, as the case may 

 be, and therefore varying but slowly, and through but a small entire range. 



" (2) The lunar or solar diurnal tide varies as a simple harmonic function 

 of the hour-angle of period 360°, or twenty-four hours, with an amplitude 

 varying always in simple proportion to the sine of twice the declination of the 

 disturbing body, and therefore changing from positive maximum to negative, 

 and back to positive maximum again, in the tropical* period of either body 

 in its orbit. 



* Tlie tropical period differs from the sidereal period in being reckoned from the first 

 point of Aries instead of from a line fixed in space, the difference for the case of the sun 

 being only one in 26, WO years, and for the case of the moon one in 13 X 18-6. 



