(b) expanding (1 - ca + a^) ' by the binomial theorem to obtain 



l-|[-2c<..«^] .l.|.|[-2c<..»^]^ 



(c) collecting powers of a to obtain 



1 + 3ca + I [Sc^ - l] a2 - | [tc^ - 3c] a^ + . . . (13) 



Since a = a/d < 1.67 X 10" , terms in a^ may be neglected in comparison 

 with a. Substituting the above expression into (a) and neglecting terms in a^ yield: 



Collecting terms in a/d yields equation (9). Trignometric identities may be used to convert 

 the products of trignometric functions into sums of trignometric functions, with arguments 

 which are sums and differences of the original arguments, to obtain 



3.M 

 2E 



("1) [sin 2z + j{5 cos2 z - l)Jsin z (15) 



Note that the tide-generating force is the horizontal component of the difference 

 between the gravitational force of the Moon or Sun at the center of tlie Earth and the 

 gravitational force of the same body at the surface of the Earth. Equations (10) and (15) 

 show that tliis force is inversely proportional to the cube of the distance between the Earth 

 and the other heavenly body. This explains why the Earth is affected more by the 

 gravitational force of the Sun than the gravitational force of the Moon, but is affected by 

 the tide-generating force of the Moon more than by the tide-generating force of the Sun. No 

 other astronomical body exerts a significant tide-generating force on the Earth. 



The distances between the Earth and Sun and Moon are periodic variables. By expressing 

 these in the form 



d = d 1 + 



29 



(i) 



