188 



Mr. G. H. Darwin on 



[Dec. 19, 



the increase of distance comes an increase of periodic time round the 

 earth. 



This general explanation remains a fair representation of the state 

 of the case so long as the different harmonic constituents of the aggre- 

 gate tide-wave do not suffer very different amounts of retardations ; 

 and this is the case so long as the viscosity is not great. 



Prop. II. The attraction of the moon on a lagging fortnightly tide causes 

 the earth's obliquity to diminish, but does not affect the diurnal rota- 

 tion ; the reaction on the moon causes a diminution of her distance, 

 and periodic time. 



The fortnightly tide of a perfectly fluid earth is a periodic increase 

 and diminution of the ellipticity of figure ; the increment of ellip- 

 ticity varies as the square of the sine of the obliquity of the equator 

 to the ecliptic, and as the cosine of twice the moon's longitude from 

 her node. Thus the ellipticity is greatest when the moon is in her 

 nodes, and least when she is 90° removed from them. 



In a lagging fortnightly tide the ellipticity is greatest some time 

 after the moon has passed the nodes, and least an equal time after she 

 has passed the point 90° removed from them. 



The effects of this alteration of shape may be obtained by substi- 

 tuting for these variations of ellipticity two attractive or repulsive 

 particles, one at the North Pole and the other at the South Pole of the 

 earth. These particles must be supposed to wax and wane, so that 

 when the real ellipticity of figure is greatest they have their maximum 

 repulsive power, and when least they have their maximum attractive 

 power ; and their positive and negative maxima are equal. 



We will now take the extreme case when the obliquity is 90° ; this 

 makes the fortnightly tide as large as possible. 



Let the plane of the paper be that of the ecliptic, and let the outer 

 semicircle be the moon's orbit, which she describes in the direction of 



