22 Prof. Challis on a Mathematical Theory of Tides. 



in the problem of the tides. In order to simplify the reasoning, 

 it will be supposed (1) that the solid part of the earth is sphe- 

 rical ; (2) that this part is covered by water, the depth of which 

 is small compared with the earth's radius ; (3) that the moon 

 revolves about the earth in the plane of the equator at her mean 

 distance with the mean angular velocity. We may, if we please, 

 abstract from the earth's rotation by conceiving an equal angular 

 velocity to be impressed on the moon in the opposite direction. 

 Conceiving, therefore, the earth to have no motion, and taking 

 its centre for the origin of rectangular coordinates, let the axis of 

 z coincide with its polar axis, and the axis of x pass through the 

 meridian of Greenwich. Then if \ be the north latitude, and 6 

 the west longitude, of any particle of the fluid distant by r from 

 the origin, we shall have 



a?=rcos\cos 0, z/=rcos\sin#, z = r sin\. 



If G be the usual measure of gravity at the upper surface of the 

 fluid at a certain distance a from the centre of the spherical mass, 

 the force at any point in the fluid at the distance r will, on ac- 



(a — v\ 

 1 + A 1 



whether the mass be homogeneous or not. The factor k depends 

 on the ratio of the density of the water, or generally of the su- 

 perficial strata, to the earth's mean density, and may readily be 

 calculated if that ratio be given. For instance, I have found 

 that if the ratio be one-sixth, ^ = | ; if one-third, £=1. 



Again, let m be the moon's attraction at the unit of distance, 

 referred to the same unit of measure as G, R the mean distance, 

 and yut her angular distance westward from the meridian of 

 Greenwich at the time t reckoned from the meridian transit, 

 the apparent angular velocity /ul being equal to the excess of the 

 earth's rate of rotation above the moon's mean motion. Then, 

 the centrifugal force due to the earth's rotation being o> 2 at the 

 unit of distance from the axis, the usual investigation to the first 

 power of the ratio of r to E gives 



X=-G(l+* 5Lzr)^ + © 2 ^+^ 8 ^(3cosV-l)+ ysin2/A 

 Y= -G(l +k fl=r)? + „»y+ ^(y(3sinV-l) + Y sin2 /4 



Hence 'K.dx + Ydy + Zdz is in this instance an exact differential, 

 and (fi?F) may be substituted for it. Hence also we may put 



