Placing, for convenience, r/R=p: 



RyD^=(l-2p cos 9+p^)-^''' 

 = [l-p{2 cos d-p)]-^'K 



Expanding the second member into the binomial series: 



RyD' = l-\-3/2 p(2 cos e-p) ^15/8 p\2 cos d-py-\- ■ ■ ■ 

 = 1+3^ cos d—S/2 p^ll — 5 cos^ d)+ terms in the 

 cubes and higher powers of ^. 



Since the distance from the moon to the earth is approximately 60 

 times the earth's radius, the cubes and higher powers of p—rjR have 

 values of 1/216,000 or less, and the terms containing them are too 

 small to be considered. Substituting, in equation (3), the expression 

 derived for R^/D^, reducing and again dropping the cubes of p: 



jr=Mix [p(3 cos^ ^-l) + 3/2 p2(5 cos^ 0-3 cos e)]IR'" 



=Mix (r/R^) C3 cos2 0-1) +3/2 AMr^R^) (5 cos^ 0-3 cos 0). (4) 



The numerical value of the coefficient of the second term of equa- 

 tion (4) is 3r/2R times, or in the order of l/40th or less of, the numerical 

 value of the coefficient of the first term. For the accuracy in general 

 necessary, the second term may be disregarded, giving: 



fr=MfM{rJR') (3 cos^ 0-1). (5) 



The distance of the moon from the earth is astronomically measured 

 by its parallax, which may be defined as the angle subtended by the 

 radius of the earth at the distance of the moon. The parallax varies 

 as the reciprocal of the distance, or as 1/R. Since the second term of 

 equation (4) contains 1/R to the fourth power, it is called the term 

 dependent on the fourth power oj the inoon's parallax. 



16. The component of the lunar differential attraction in the direc- 

 tion perpendicular to CP, in the plane CPO, is similarly: 



ih= (Mn/D') sin P- (Mfx/R^) sin 

 from figure 1: 



Z>sin P=OA=R sin 0, 

 giving: 



sin P=R sin 0/Z>, 

 so that: 



fh=Mfi{R sin d/D^-sin d/R') 

 =MfjL sin d{RyD^-l)IR'-. 



