Gravitational Attraction on Chemical Composition. 255 



some fixed direction in the plane of the moon's orbit. At 

 time t let the radius-vector drawn from the sun's centre to G 

 make with GD an angle <£, and let the line earth-to-moon 

 make with GL an angle 6 at the same instant. We assume 



=(ot + $, where $ is small, j 



<f> = £lt + j3, where ft is constant, J ' 



& and O being constant angular velocities. Let r, the distance 

 from earth to moon, = a + p, where a is constant and p a very 

 small periodic term. The radial acceleration of the moon 

 relatively to the earth is 



'-'**=-£ +&«»(*-*), ... (7) 



where A/a 2 = o> 2 a. In this equation, substitute for r, 0, <£, 

 rejecting squares and products of jn, p, 3; we thus obtain 



'p-Z(o 2 p-2awk= / A-R-*cos{((D-n)t-(3' ( . . . (8) 



Similarly for the perpendicular component acceleration, in the 

 direction of increasing, we have 



2rd + r#=-/*K- 8 sin (#-</>), 



which to our order of approximation is the same as 



2cop + ad==- fJ LR- 2 sm{{cD-Q,)t-j3}. . . (9) 



7. We are only concerned here with that particular 

 integral of the equations (8), (9) which is directly dependent 

 on the value of yu., and which makes p and 3 strictly periodic 

 functions of the time, the period being evidently 2ir j '(© — O) , 

 or one lunar month. Thus, for example, when (yj is inte- 

 grated with respect to the time, no constant of integration 

 has to be introduced, and the values of p, S are readily 

 obtained. That for $ is 



» = -5C n(2 m -fl)( a ,-fl)s sm{(m - n >- /3 } 

 = -n^- g( 2 g -ix g -i) ' 8m/ >- Q)< -^ ; - (10) 



if we put yi~ 7 2 = «G, and q^co/Q, the number of lunar 

 months in a year. Here MG 2 /R 2 is the strength of the sun's 

 gravitative field at the earth-moon system, and is thus equal 

 to n 2 R, so that finally (10) becomes 



•--'•r d&^ir * 1 ^- *-*- (11) 



