199] EQUILIBRIUM THEORY OF TIDES. 537 



This equation shows that the tidal surface is a prolate ellipsoid of 

 revolution, with its axis pointing at the disturbing body. 



Let us now express cos Z in terms of the latitude iff of the point 

 of observation and of the declination d and hour -angle H of the 

 disturbing body, which for brevity we shall call the moon. If we 

 take axes in the earth as usual, with the XZ- plane passing through 

 the point of observation and measure H from this plane, we have 

 for its coordinates and those of the moon respectively 



r cos iff, D cos d cos H, 



; D cos d sin H, 



r sin ijt, D sin d , 



from which we obtain the cosine of the angle included by their radii 

 cos Z = cos iff cos d cos H -f sin if> sin d. 



Squaring this, replacing cos 2 H by -_ - (1 4- cos 2-ET), cos 2 iff cos 2 d by 

 (1 sin 2 i{i)([ sin 2 d), we easily obtain 



3 cos 2 Z 1 = -~ [cos 2 d cos 2 ty cos 2H + sin 2 d sin 2^cosH 



(l-3sin 2 (?)(l-3sin 2 ij/n 

 8^ J- 



Inserting this in 171), replacing g by its approximate value - Tr and, 



as we have already done, neglecting the attraction of the disturbed 

 water, we have the equation for the tide, 



172) * = ^^ [cos 2 d cos 2 ^ cos 2H + sin 2 d sin 2 # cos H 



(1-3 sin 2 d) (1 - 3 sin 2 i|;)-| 

 ~~3~~ J 



The first term in the brackets, containing the factor cos 2 JET, 

 where H is the moon's hour -angle at the point of the earth in 

 question, is periodic in one -half a lunar day, consequently this term 

 has a maximum when the moon is on the meridian, both above and 

 below, low water when the moon is rising or setting. The effect of 

 this term is the semi-diurnal tide, which is the most familiar, with 

 two high and two low waters each day. This tide is a maximum 

 for points on the equator, where cos 2 ty = 1, and for those times of 

 the month when cos 2 d = 1, that is when the moon is crossing the 

 equator. These are the so-called equinoctial tides. 



The second term, containing the factor cosJS, is periodic in a 

 lunar day, and gives the diurnal tide. This gives high water under 

 the moon, and low water on the opposite side of the earth. On the 

 side toward the moon, these two tides are therefore added, while on 



