26 TJ. S. COAST AND GEODETIC SUEVEY. 



Substituting the value of r from (41), 



(-IT 



tan lAi = 3/2 ^^ sin 2Q 



1 ^^3 — - (3 cos2 e -1) 



(56) 



ni (ft -f- 7/, )^ 



The values of - and — -^^ — are each very small compared with 



unity, and the value of the bracketed portion of (56) is therefore a 

 very close approximation to unity. We may therefore write 



tan yp^ = 2^^ sm 2d (57) 



as the tangent of the angle between the radius vector and the result- 

 ant of the forces at the point P. Comparing this with (54) , we find 

 it to be the same as the angle made by the normal with the radius 

 vector, indicating that the resultant force is normal to the surface 

 at any point P. 



If we let d represent the mean distance of the moon and substitute 

 the numerical values from Table 2 for the coefficient in (54) we 

 obtain 



tan 1^ = 0.000,000,084 sin 2 d (58) 



in which i// as a maximum value of about 0.017" when = 45°. The 

 maximum deflection of the normal due to the tide-producing force of 

 the sun is about 0.46 times as great, or 0.008'', approximately. 



Let us now consider the disturbance in a spherical surface due to 

 the potential depending upon the fourth power of the moon's parallax 

 (35). This potential will become zero when 0=90°, and also when 

 cos 0= V3/5. Letting a represent the radius vector at either of these 

 points, we have as the equation for the equilibrium surface due to 

 the potentials (32) and (35) 



^ +i '^ (5 C0S3 0-3 cos B) =^ (59) 



r ^ d^ ^ a 



from which may be obtained 



i^|^(5cos3 0-3cos0)=^^(r-a) (60) 



r = a + u', (61) 



Letting 

 we have 



J -g J. (5 COS' 9 - 3 COS « - j;^^:^, = - - 4J-) + 1 j- 1 - etc. (62) 



Neglecting powers of — above the first, 



— = i ^j, (5 cos^ e - 3 cos 6) (63) 



