﻿of Tidal Action. 



221 



represent the masses of the earth and the moon, r the equatorial 

 radius of the former, D the distance between the centres of both 

 bodies, A the difference of terrestrial longitude between the crest 

 of the tide-wave and that place where the moon is vertical, the 

 difference of longitude between the crest of the wave and any 

 point in the channel ; and let y be the variable height of the 

 tide at this point ; while the breadth of the channel is denoted 

 by b, and is supposed to be small in comparison to the earth's 

 dimensions. Then brd$ will express the magnitude of the infi- 

 nitesimal portion of the water which rises above the proper geo- 

 metrical boundary of the terrestrial spheroid, aud its attractive 

 force on the moon will be 



tfgbyrdfy , r . 



l) 2 -2Drcos(A-</))-fr 2 cos 2 (A-(/>)' * * *> * 

 in which g measures the attraction exerted at the distance k by 

 a volume of water assumed as the unit of matter and having a 

 spherical form. Of the force represented by formula (5), the 

 component acting horizontally and tending to change the direc- 

 tion of terrestrial gravity on the moon will be 



k*bgyr 2 sin (A- $)d<j> 



(D 2 -2Drcos(A-</>)+r 2 cos 2 (A-<£))#' l 



Making this expression equal to df, transforming it into a series 

 of which it is necessary to retain only the first two terms, and 

 putting C for k 2 gb, there results 



? Cyr a sin(A-cft)ffi 3Cyr 3 sin2(A- 



(p)dcf> 



(7) 



df- D 3 - 2 D 4 



The value of the force /which occasions the secular change in 

 the lunar movement may be obtained by integrating the last 

 equation ; but for this purpose y must be expressed in terms of 

 <jb ; and in ascordance with the theory of the tides and the laws 

 of periodicity in their movements, y may be assumed equal to 

 h cos 2(/> + A' sin 2</> ; h and h' being two constant quantities, the 

 latter small in comparison to the first, and depending on the 

 effect of friction. Equation (7) becomes, on the substitution of 

 this value of y , 



C/ir 2 sin (A + </>)#_ C/zr 2 sin (30-A)# 

 aj — 



+ 



+ 



2D 3 

 3CAr 3 sin 2A<ty 



4D 4 



Cflr'co s (A + </>)# 



2D 3 



3CAV 3 cos 2Acty 



~ 4D 4 



2D 3 



3C^ 3 sin(4 (fr- 

 4D 4 



2A)d<j> 



C//r 2 cos(3c/>-A)^ 



2D 3 

 3C/Vr 2 cos(4(/)-2A)^ 



4D 4 



(8) 



