﻿222 Mr. D. Vaughan on the Secular Effects 



Now, on integration within the limits of (f> = and (£ = 27T, the 

 terms containing this variable angle disappear ; and since A is 

 constant, there results 



/= ^^ {h sin 2 A 4 ti cos 2 A) ... . (9) 



If the waters of our globe, instead of being confined to the 

 vicinity of the equator, were made to occupy a number of regular 

 channels ranging with the parallels of latitude, and if br denote 

 the breadth of one of these channels, the polar distance of its 

 middle part, while the notation already given is retained for the 

 remaining items, the tidal swelling of the fluid confined under 

 the given parallel will have its tangential action on the moon ex- 

 pressed by 



3^Wsia3q (As . n2A + ^, cos2A) _ _ (lQ) 



This may be easily found by using in the foregoing investigation 

 r sin 0, the radius of the parallel of latitude, instead of r the 

 radius at the equator. Now instead of the complicated case 

 which our terraqueous world presents, I shall, like most writers 

 on the tides, take an equivalent one in which the entire globe is 

 supposed to be covered with water having a depth equal in all 

 places, or varying regularly with the latitude according to some 

 obvious law. We may suppose this hypothetical ocean divided 

 into watery zones or canals by partitions parallel to the equator ; 

 as the number of these divisions become infinite, the breadth of 

 each will be represented by rdO, and its tangential force on the 

 moon by d¥. From formula (10) there is thus obtained 



,-p, ZirW-qr^ sin 3 0^0 ., . _. 7 , rt . . ,__, 

 2D* (Asin2A + A , cos2A). (11) 



The angle A may without much error be regarded as constant 

 for all latitudes ; but it is proper to consider the greatest height 

 of the tide-wave as depending on the distance from the equator; 

 and supposing it proportional to the cosine of the latitude, we 

 must substitute for h and h' in the last equation h sin O and 

 A' sin 0. Making this substitution and integrating within the 

 limits of = and = 90°, we obtain 



F = 9 ^l r * (h sin 2 A + hi cos 2 A) . . . (12) 



Some idea of the small effect of the tides on lunar motion may 

 be derived from a numerical estimate of the value of F in the 

 last equation, supposing the angle A to be 45°, and the greatest 

 height of the tides equal to three feet. For this purpose it will 

 be most convenient to take for the unit of attracting matter a 



