﻿of Tidal Action. 219 



sists between the changes which the spheroidal form of the earth 

 occasions in lunar and terrestrial motion. 



It is easy to estimate with more accuracy than is common to 

 this department of science the ratio between the minute errors 

 of the earth and the moon in recording mean time, and thus to 

 derive important information respecting the portion of these 

 errors discoverable by observation. As the moon, while only ^ 

 of the earth in respect to mass, moves in its orbit with about l\ 

 times the diurnal velocity of our equator, our planet, supposing 

 it to be homogeneous, would derive from its rotation about fifteen 

 times the momentum which the moon owes to her orbital motion. 

 Now, the action to which each body is exposed from tidal move- 

 ments being in the ratio of sixty to one, our globe must lose four 

 infinitesimal parts of its angular velocity from the disturbance 

 while the moon gained one, the mean distance between both 

 orbs being regarded as immutable. But it is well known that 

 if the moon's velocity were increased 1 per cent., there would be 

 an augmentation of 2 per cent, in the size of her orbit, and of 3 

 per cent, in the time of her periodical revolution. From this it 

 follows that the relative change in the mean motion of our satel- 

 lite, from the occurrence of the tides, is about three-fourths of 

 that which the length of the day must experience from the same 

 cause, and that only one-fourth of the permanent change thus 

 occasioned in the rotation of our planet would be revealed by a 

 comparison of ancient and modern eclipses. 



This result, however, requires some corrections, not only for 

 the inequalities of density in terrestrial matter, but also for the 

 ever varying declination of the moon. To correct for declina- 

 tion, put L for the moon's longitude, I for the inclination of its 

 orbit to the plane of the equator, M for the mass of the earth, 

 and K for its radius of gyration, and &w for the amount of the 

 secular change in its angular motion in a given time. Then in 



equation (2) substituting — -j- — for /' in conformity with the 



principles of dynamics, and for cos s its equal a/1 — sin 2 Lsin 2 !, 

 extracting the square root of the latter in a series and reducing, 

 we obtain 



MK 2 & B,//. sin' 2 1 sin 2 Icos2L Q \ 



Now if L be expressed in terms of the time /, and I for the pre- 

 sent be regarded as constant, the terms containing L will disap- 

 pear on integration, and the last equation will yield the following: 



MK 2 Sa>=^(l- S -^J nearly. . . (4) 

 If the moon moved in the plane of the ecliptic, this formula 



