HARMONIC ANALYSIS AND PREDICTION OF TIDES. 27 



or 



u' = ^^^{5 cos^ 9-3 cosd)a (64) 



as the equilibrium height of the tide due to that part of the lunar 

 force depending upon the fourth power of the moon's parallax. 



In forming equation (59), a was taken at the radius vector along 

 which the potential due to the fourth power of the moon's parallax 

 was zero. To determine whether this is the mean radius, we find 

 the volume included in the surface represented by (64) . 



J27r /»7r ra+n' 

 I 7^ sin 6 d(j> de dr 

 J oj 



(65) 



J27r r-T 

 {a + uy sin dd<l>dd 

 o J o 



From (64) 



r J/a* "I 



{a + uT = a^ l+3/2-^^(5cos3^-3 cos0)+etc. 



(66) 



the terms containing- the powers of ^ above the fourth being neglected. 

 Substituting (66) in (65) 



1+3/2 -^j, (5 cos^ d-3 cos 6) \sindd<t>dd 



= 2/3^3 pV = 4/37ra3 ^^^^ 



As (67) is the volume of a sphere with radius a, it is evident that 

 this is the mean radius of the volume included in the surface repre- 

 sented by (64), and that u' is the amount of the disturbance in the 

 mean surface due to the force depending upon the fourth power of 

 the moon's parallax. 



Letting \p' equal the angle between the radius vector and normal 

 to any point P in this surface, 



i^nr--^, (68) 



and from (61) and (63) 



if a* 



r = a + i -gr j4 (5 cos^ ^-3 cos ^) a (69) 



Then 



' J=-i^^(15cos2^-3) sine (70) 



Substituting (69) and '(70) in (68) and neglecting powers of -? 

 above the fourth 



tan yp' = 3/2 ^ j, {5 cos^ ^- 1) sin (71) 



72934^24t- 3 



