280 Prof. Challis on two new Theorems 



Hence, if d(f> be the small augle described by the radius-vector 

 r, corresponding to the increment d$ of longitude in the time df, 

 and if c{- + c^ + c/ — k'^, and p-n be put for Cy in the small term, 

 \vc have to the same approximation, 



'■i|!=^+«^e-'cos2«. (B) 



But 



_.„ dr^ r^dAr 



~ df- ^ df 

 Hence, by the equation (A), 



dr"^ h^ 2fj, m'r^ ^_ ,p> 



^+;:^-7--2^3 + C-0 (C) 



This is the equation that would be obtained if a body were sup- 

 posed to be acted upon by the central force -^ — -^-jry Hence this 



equation proves Theorem I. 



It may be remarked, that in the foregoing reasoning the incli- 

 nation of the orbit to the plane of the ecliptic is taken into 

 account, and that there is no limitation of the value of r except- 

 ing that its ratio to a' is a small quantity of the second ordei-, 

 the approximation in other respects proceeding solely on the 

 hypothesis that the true longitudes of the sun and moon differ 

 by small quantities from mean longitudes. 



But it is evident from theoretical considerations, that on that 

 hypothesis the moon's orbit cannot differ much from a circle, 

 and it is known from common observation of her apparent dia- 

 meter that this must be the case. Our approximation will con- 

 sequently conform to the moon's motion, if the radius-vector be 

 supposed to differ by a small quantity of the first order from a 

 mean value a. If, therefore, a be put for r in the small term of 

 the equation (C), it would seem that all small quantities of the 

 second order are taken into account, and that we have to inte- 

 grate the equation 



dr"^ h^ 2fM ni'a ^ ^ 



This integral would give the same relation between r and t as 



that in an orbit described by the central force ^, the constant h 



being twice the area described by the radius-vector in the 

 unit of time. And the equation (B) shows that h is also the 

 me(m value of twice the area described by the moon's radius- 

 vector in the unit of time. Hence we must conclude that the 

 mean distance and mean periodic time in this approximation to 



