202 Prof. Challis on the Theory of the Moon's Motion. 



to such terms, and to indicate processes of integration which are 

 appropriate to the lunar problem. 



It will suffice for this purpose to suppose the force to be wholly 



central, and to be equal to fiu?——, u being the reciprocal of 



the radius- vector. Thus we have the known equation, 



w +u ~i? + h=° M 



Multiplying by 2 -^ and integrating/ 



dit 2 a 2/xu /j/ r 



__ + ,-____ +c= o. 



Hence, making -^=0, the equation for determining the apsidal 



distances is 



4 2/xm 3 p 2 _/// , 



1? + h 9 [ ' 



Now, since C is positive for an elliptic orbit, or one approaching 

 to an ellipse, if this equation contains two positive roots, it 

 must, by the theory of equations, contain a third positive root, 

 because the last term is negative. Hence there must be a third 

 apsidal distance in addition to the two belonging to the approx- 

 imate ellipse. The third apse is accounted for by considering 

 that if the radius-vector be very large, and consequently u very 



small, the repulsive part of the force, viz. — — , may exceed the 



attractive part /xu 2 , even when /J is supposed to be small. 

 Hence the analysis embraces a separate curve, containing in- 

 finite branches and one apsidal distance, in addition to the 

 eccentric orbit. If, therefore, it be required to integrate the 

 equation (1) on the supposition that the orbit is of small 

 eccentricity and approximately elliptical, some method must be 

 adopted which will exclude the third apsidal distance. One 

 method proper for this purpose is that indicated by Mr. Airy 

 (Mathematical Tracts, 3rd edition, Art. 44* of the Lunar 

 Theory, p. 32), which consists in substituting b + w for u, and 

 neglecting powers of w above the first. This is virtually the 

 same as the method employed in Pratt's Mechanical Philosophy 

 (Art. 334, p. 300), where b + u—b is substituted for u, and 

 powers of u—b above the first are omitted. Making the latter 

 substitution, the equation (1) becomes, 



^2 + V h%V h* + h*P ' 



