264 Prof. Challis on the Mathematical Theory 



of the disturbing: force than the first, and the small term invol- 

 ving k, will be omitted. In that case Q= ^, and the equation 

 (/3) becomes 

 j0== _Gr + |^(3cos 2 Xcos 2 ((9-^)-l)--^ + ^W- ft) 



Now the expression for -—■ shows that the vertical velocity is 



rrr 



always zero where \— ± -^-, and consequently that at the two 



poles the radius r of the surface is constant. Let a be this value 

 of the radius, and G be the attraction of gravity at these posi- 

 tions. Hence, since ~ also vanishes where X= + -^ , if ^ be 

 at <v 



the atmospheric pressure, which is supposed to be uniform and 

 constant, we shall have 



This equation gives the value of -*]r(t), by substituting which in 

 the equation (7), and putting for -jj its value 



-2ftc(r 2 + 2 ^) cos 2 \cos2(0-/^), 

 the following result will be obtained : 



+ (§S +3/tC (? + |5)) cos2xcos2 ^-^)- 



The equation of the exterior surface of the ocean results from 

 this equation by putting -nr for p. It will thus be found, omit- 

 ting terms containing m 2 &c, that 



3m« 2 „ . 

 r=a+ 4B?G C0S X \ 



+ llPS + -5- ( a + ©^"'(M-J 



If we give to p any arbitrary value jo p the equation (8) will 

 apply to a varying surface at all points of which the pressure is 

 p l at any time t. By obtaining (Sp^, that is, by making p x 

 vary with respect to coordinates only, we should get the change 

 of p x in passing from one surface of equal pressure to a conti- 



(8) 



