26 Prof. Challis on a Mathematical Theory of Tides. 



tion of the problem of the resistance of the air to the motion of 

 a ball-pendulum. 



This point being allowed, the arbitrary constant C will enable 

 us to introduce the condition that the inferior limit of the water 

 is a spherical surface of constant radius. For, putting b for this 

 radius, we shall thus have u' = where r=b } and consequently 



a + qb -£» 

 C= — f-e a. 



After the substitution of this value of C in the expressions for 

 ft, u', v', w ! , the functions of r in brackets may be expanded in 

 terms proceeding according to the powers of the small quantity 

 q, which is of the order of the ratio of the centrifugal force at 

 the earth's equator to the force of gravity. When this is done 

 and terms wholly insignificant are omitted, the results are 



*' =_ Sk{ b *~£ (r-6) 9 (r + 2i)) cos* \ sin 2(0-^), 



«'= ^(r*-^) cos* X sin 3(0-^), 



W - SfTr( l - St* (r-bnr + 2b)ym2X S m2(e-t>t). 



These approximate values of u', v\ w' satisfy the equality 

 (d$) = udr + v'r' cos Xdd + w'rd\. 



It will be seen that v! = at the points for which \= + -, and 



that consequently the radius vector of the surface is at these 

 points invariable. We may therefore suppose the distance a 

 to be this value of the radius vector, so that G is the force of 

 gravity at the two poles. 



Resuming the equation (77), and substituting in it for ~- from 



the above value of <£', the resulting equation for calculating p is 



p - _ Gr + HJ (3 cos 2 X cos 2 (0-fit) - 1) 

 ~5(^-|^(^^)V + ^))cos 2 Xcos2(^^)+fW. 



