r = a + 



of Oceanic Tides. 271 



We have thus determined the value of c l and proved that it is 

 always positive. The- value of c 3 is given by the equation 



c 3 = l ', and for calculating c q we have 



2fju 2 c q _ 3ma/jub Sma^b 



~Go7 " "~ WWb '' or °* ~ 8R/p~ ' 



By employing the value now obtained for -— -, we shall have, 



since 



c } af b 5 \ 3ma 2 2mc x ( - c 2 2b 5 \ 



for the equation of the upper surface of the water 



^3^cos 2 \+— -f l-^Jcos 2 \cos2((9-^) = « + 



So far as the first and last terms of the second form of the equa- 

 tion indicate, the surface, since c x is positive, is that of a prolate 

 spheroid, the axis of which is always directed to the place of the 

 attracting body. The other term gives the amount of deviation 

 from this form. 



The first form of the equation shows that there is a constant 

 elevation of the waters, varying when the body is in the equator, 

 as the square of the cosine of latitude, and that the maximum 

 variation from this elevation, or difference between high and low 

 water at a place whose latitude is X, is 



_J_^l__J cos2X . 



This formula may be tested by applying it to the case in which 

 fjb=0, that is, the case of equilibrium of the waters, for which, 

 as is known, 



r = a+ r R3 cos 2 X cos 2 (6— /.it). 



Now this equation results from the second of the above expres- 

 sions for r by supposing that 



Cjaf ^\_Sma^ 



7TV «V"~4GR 3J 

 whence it follows that in the case of equilibrium the difference 

 between the greatest and least values of r is 9rT » 3 cos 2 X,, as is 



