1878.] the Precession of a Viscous Spheroid, §-c. 191 



produce a ridge of elevation all along the neighbourhood of the arc 

 LRL', together with a compression in the direction of an axis perpen- 

 dicular to the plane of the ring. This tidal spheroid may be conceived 

 to be replaced by a repulsive particle placed at P, the pole of the ring, 

 and an equal repulsive particle at its antipodes, which is not shown in 

 the figure. 



Now suppose that the spheroid is viscous, and that the tide lags ; 

 then since the planet rotates in the direction of the curved arrow at S, 

 the repulsive particle is carried past its place, P, to P'. The angle PSP' 

 is a measure of the lagging of the tide. 



We now have to consider the effect of the repulsion of the ring on a 

 particle which is instantaneously and rigidly connected with the planet 

 at P'. 



Since P' is nearer to the half L of the ring, than to the half L', the 

 general effect of the repulsion must be a force somewhere in the direc- 

 tion P'P. 



Now this force P'P must cause a couple in the direction of the 

 curved arrows K, K' about an axis, KK', perpendicular to LL', the 

 nodes of the ring. The effects of this couple, when compounded with 

 the planet's rotation, is to cause the pole S to recede from the ring 

 LRL'. Hence the inclination of the planet's equator to the ring 

 diminishes. 



Secondly, the force P'P produces a couple about S, adverse to the 

 planet's rotation about its axis S. If the obliquity of the ring be 

 small, this couple will be small, because P' will lie close to S. 



Lastly, it may be shown analytically that the tangential force on 

 the ring in the direction of the planet's rotation, corresponding with 

 the tidal friction, is exactly counterbalanced by a tangential force in 

 the opposite direction, corresponding with the change of the obliquity. 

 Thus the diameter of the ring remains constant. It would not be 

 very easy to prove this from general considerations. 



It may be shown that, as far as concerns their joint action, the sun 

 and moon may be conceived to be replaced by a pair of rings, and 

 these rings may be replaced by a single one ; hence the above propo- 

 sition is also applicable to the explanation of the joint action of the 

 two bodies on the earth, and numerical calculation shows that these 

 joint effects exercise a very important influence on the rate of variation 

 of obliquity. 



To return to the paper : the retardation of the earth's rotation would 

 cause an apparent acceleration of the moon, if that body were unaffected, 

 but this is partly counterbalanced by the true retardation of the moon. 

 We thus have the means of connecting an apparent acceleration of the 

 moon with the heights and retardations of the several bodily tides. I 

 have applied this idea to the supposition that the moon is subject to an 



