﻿218 Mr. D. Vaughan on the Secular Effects 



can be depended upon for correcting the time kept by the other, is 

 not difficult if the moon's orbit is supposed to be an exact circle. 

 The centrifugal forces arising from the movement of the earth and 

 the moon around the common centre of gravity between them, 

 must have a resultant in a line passing through the centre of gra- 

 vity of our planet. In the absence of tides, the resultant of the 

 attraction between the two great bodies would, in its mean posi- 

 tion, meet the earth's axis which passes through the same centre ; 

 for though it may be caused to deviate from this position by the 

 influence of mountains or by irregularities in the density of ter- 

 restrial strata, the deviation would take place to an equal extent 

 on opposite sides of the axis, and would have its effect evenly 

 compensated in the course of every period of rotation. But the 

 influence of the swollen tidal waters causes the attractive force 

 of the earth on the moon to act in a line passing a little east of 

 the terrestrial axis; and it is on this slight deviation from the 

 axis that the permanent change of motion in both orbs depends. 

 Let D denote the distance between the centres of the earth and 

 the moon, R the measure of the attraction between both bodies, 

 and / the shortest distance between the earth's axis and the line 

 which marks the direction of this attractive force; then R may 

 be resolved into three components — one coinciding in direction 

 with centrifugal force which it balances, a second depending on 

 the ellipticity of our planet, and serving to produce the preces- 

 sion of the equinoxes and the nutation of the earth's axis, while 

 the third, much smaller in magnitude, depends on the attraction 

 of the tidal waters on the moon. If the last component (which 

 acts in the direction of lunar motion) be represented by/, then 

 from the principles of the resolution of forces it will appear that 



/r,S V 



From the theory of rotation it may be also easily inferred that if 

 f denote the force exerted in changing the earth's rotation by 

 the action of R, then 



j,/ R/cos s ,~v 



where r stands for the earth's equatorial radius, and s for the 



moon's declination. 



D cos s 

 From these expressions it follows that/' is equal to/ ; 



and the changes of momentum resulting from / and J' in the 

 same time, being proportional to the forces themselves, must have 

 one to the other the ratio of r to D cos s. In this we may 

 observe a conformity to the principle of the preservation of areas 

 and moments. Laplace has shown that a similar relation sub- 



