262 Prof. Challis on the Mathematical Theory 



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

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

 total attraction of the solid and fluid portions towards the centre 

 at any point within the fluid at the distance r will be very nearly 



G( 1 -f- k ), k being a certain constant depending on the ratio 



of the density of the water to the earth's mean density. If this 

 ratio be one-sixth, which is known to be about its actual value, 

 the calculation of the attraction shows that # = 1 nearly. 



Let m be the attraction of the disturbing body at the unit of 

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

 distance, fit its angular distance westward from the meridian of 

 Greenwich at the time t reckoned from Greenwich meridian 

 transit, fi being the excess of the earth's rate of rotation above 

 the body's mean motion. Then, omitting powers of the ratio of 

 r to R above the first, the usual investigation gives 



X = -G(l + *£=:)? + p(*(S cosV-1) + I sin *tfj, 

 Y = -G(l + k ~) y ~+ |e(y(8 sinV-1) + y m ty*) 



Hence in this case JLdx + Ydy + Ttdz is an exact differential. 

 Substituting (d¥) for it, we have, by integration, 



171 



+ jjjgs (# 2 (3 cos 2 fit— 1) + 2/ 2 (3 sin 2 /it- l) + 3wy sin Zfit-z*). 



Since (d¥) — (dp) is equal to f^Jdx + C^dy + ( J") dz > au( * 



this quantity is consequently an exact differential, if we repre- 

 sent it by (dQ), we shall get by integration p = ¥—Q + yfr(t). 

 Hence, substituting for x, y, z in F their values in polar coordi- 

 nates, it will be found that 



-<:^ \ . m 



+ ^( 3cos ^ cos2 ^-^)-i)-Q+fW.J 



The investigation in the January Number led to an expression 

 for <£, which, as already stated, did not satisfy the equation («) ; 

 and although it gave results in accordance with observed laws 



