196 Dr. C. Cliree on Applications of Physics 



The angle through which the radius-vector is rotated from 

 its undisturbed position, in the direction away from = 0, is 

 equal to v a /a } and so is known from (61). As v a /a is nega- 

 tive for all values of between and 7r/2, this rotation is 

 really towards the moon at every point of the illuminated 

 hemisphere. 



§ 29. We next require to find the inclination of the re- 

 sultant force to the radius-vector over the surface. 



Employing (57) and (59) in (45) we find for the complete 

 value of the potential 



Y= -fer(r 8 /a)[l -2P 3 (M/E) (a/R) 3 (l + 5gpa/19n) 



-Hl+%Wl9rc)]. . (63) 



The component forces along and perpendicular to the 

 radius-vector are 



^ dV « IdY 

 dr 1 r dd 



Thus at a point on the surface the principal terms, which 

 alone we require, are 



R= -#, @= -3 g sin cos 0(M/E) (tf/R) 8 (l + 5gpa/19n) 



-±-{l + 2gpa/X9n), . (64) 



and the inclination of the resultant force to the radius-vector 

 is to a first approximation 



S X = S/U=3 sin 6 cos 0(M/E) (a/R) 3 (l + 5gpa/19n) 



+ (l + 2gpa/19n). . (65) 



For the apparent change of altitude, ha, in a star we have, 

 as already explained in § 23, 



Bu = S X H- va/a = 3 sin cos 0(M/E) (a/R) 3 [ 1 + ~ (gpa/n) \ 



+ (l + 2gpa/19n). . (66) 



§ 30. For the apparent change of level &/r, we require the 

 inclination of the resultant force to the normal. To obtain 

 this we may employ the result (65) in conjunction with the 

 inclination Sfa of the normal to the radius- vector, the latter 

 being given to a first approximation by 



8 * l= (- 1 %h„ = rg sin * cos 0( *P a/tC > ( M / B )(«/R) 3 



-i-{l + 2gpa/19n). (67) 



