1878.] the Precession of a Viscous Spheroid, fyc. 189 



the arrows. Let NS be the earth's axis, which lies by hypothesis in 

 the ecliptic, and let LL' be the nodes of the orbit. Let N be the 

 North Pole ; that is to say, if the earth were turned about the line LL', 

 so that N rises above the plane of the paper, the earth's rotation would 

 be in the same direction as the moon's orbitual motion. 



First consider the case where the earth is perfectly fluid, so that the 

 tides do not lag. 



Let w 2 , m 4 be points in the orbit whose longitudes are 45° and 135°; 

 and suppose that couples acting on the earth about an axis at perpen- 

 dicular to the plane of the paper are called positive when they are in 

 the direction of the curved arrow at 0. Then, when the moon is at 

 m x the particles at ~K and S have their maximum repulsion. But at 

 this instant the moon is equidistant from both, and there is no 

 couple about 0. As, however, the moon passes to m 2 there is a 

 positive couple, which vanishes when the moon is at m 2 , because the 

 particles have waned to zero. From m 2 to m 3 the couple is negative ; 

 from m 3 to m 4 positive ; and from m 4 to m 5 negative. Now, the couple 

 goes through just the same changes of magnitude, as the moon passes 

 from m x to m 2 , as it does while the moon passes from m 4 to m 5 , but in 

 the reverse order ; the like may be said of the arcs m 2 m 3 and m 3 ??2 4 . 

 Hence it follows that the average effect, as the moon passes through, 

 half its course, is nil, and therefore there can be no secular change in 

 the position of the earth's axis. 



But now consider the case when the tide lags. When the moon is 

 at nil the couple is zero, because she is equally distant from both 

 particles. The particles have not, however, reached their maximum 

 of repulsiveness ; this they do when the moon has reached Mi, and 

 they do not cease to be repulsive until the moon has reached M 2 . 

 Hence> during the description of the arc miM 2 , the couple round is 

 positive. 



Throughout the arc M 2 w 3 the couple is negative, but it vanishes 

 when the moon is at r%, because the moon and the two particles are in 

 a straight line. The particles reach their maximum of attractiveness 

 when the moon is at M 3 , and the couple continues to be positive until 

 the moon is at M 4 . 



Lastly, during the description of the arc M 4 ra 5 the couple is negative. 



But now there is no longer a balance between the arcs m-^Mo and 

 M 4 m 5 , nor between M 2 m 3 and m 3 M 4 . The arcs during which the couples 

 are positive are longer and the couples are more intense than in the 

 rest of the semi-orbit. Hence the average effect of the couples is a 

 positive couple, that is to say, in the direction of the curved arrow 

 round 0. 



It may be remarked that if the arcs miMi, ^2M 2 , m 3 M 3 , m 4 M 4 had 

 been 45°, there would have been no negative couples at all, and the 

 positive couples would merely have varied in intensity. 



