﻿220 Mr. D. Vaughan on the Secular Effects 



would indicate a reduction of nearly 4 per cent, in the estimate 

 deduced for the earth's loss of diurnal motion, on the supposition 

 that the lunar orbit coincides with the plane of the equator. 

 Had we taken the actual case, in which the moon's path has a 

 small inclination to the plane of the ecliptic, I should be regarded 

 as variable, and the investigation would be more complicated ; 

 yet it may be easily seen that the result would not materially 

 differ from that given in formula (4). 



While the inclination of the moon's orbit to the equator re- 

 duces the change in the earth's diurnal motion 4 per cent., an 

 opposite effect arises from the increasing density of the matter 

 towards the central region of our world. The law to which the 

 variable density of terrestrial strata conforms is not at present 

 sufficiently well defined to serve as a basis for an accurate mathe- 

 matical investigation ; but judging from the effects of this cause 

 on the form of the earth and the precession of the equinoxes, we 

 may safely state that it cannot add more than 12 per cent, to 

 the secular loss which the tides occasion to the motion of our 

 globe ; and accordingly a little less than one-third of this loss 

 would be made known by a comparison of the observations on 

 the moon in ancient and modern times. How far these results 

 are liable to be vitiated from the action of a resisting medium in 

 the celestial space, or from the slow contraction of our globe as 

 it constantly parts with internal heat, is a question which cannot 

 be embraced in the present inquiry. 



To obtain from the preceding equations a numerical estimate 

 of the extent to which lunar and terrestrial motions are affected 

 by tidal action, it is necessary that the value of / should be 

 known. The most reliable data for determining this value might 

 be derived by observing the small amount of deviation which 

 the position of the moon or the swelling of the tides occasions in 

 the direction of plumblines extending from the top to the bot- 

 tom of deep mines in various localities. But for obtaining such 

 an estimate from the results of observations already made on the 

 tides a different mode of investigation must be adopted. This 

 I shall now present in the most simple form, my object being 

 not so much to determine how much the length of the day or 

 the path of the moon has changed during long periods of time, 

 as to afford the means for showing the manner in which the 

 power which the tides supply comes indirectly from the immense 

 stores of force which our globe and its satellite derive from their 

 great velocities. 



I shall take first the least complicated case, in which the ter- 

 restrial waters are supposed to be confined to a uniform channel 

 encircling the earth at the equator, the plane of which I shall 

 regard as coincident with that of the lunar orbit. Let M and m 



