13.2 Prof. Challis on the Eccentricity of the Moon's Orbit. 



where R is put for — C? s + 2/u,r — A 2 . Let a(l+e) and a(l — e) be 

 apsidal distances which satisfy the equation 



-Cr 2 + 2/tr-A 2 + / i 'r 4 =0 (A) 



Since the square of yJ is not retained, it will suffice to employ 

 in those terms of the foregoing equation which involve (J, the 

 values of a and e given by the solution of the approximate equa- 

 tion, — C? s + 2/xr— A 2 = 0, that is_, to suppose the approximation 

 to commence from a fixed ellipse. Consequently, 



a 1 



a = g, ae = ^ s/ ft — h*C, 



and 



, a — Cr ,a — r 



cos -1 — - = cos -1 . 



Vft-h*C ae 



Let this arc = </>. Then the above equation between r and 

 gives to the same approximation, 



■fJT 



cos 



V ^ y 2C* V rx/ft-, 



■A 2 C 

 h*ft j _r* 2 f A 2 C - 3^ 2 ) (A 2 -/at) 3AV(/t - Cr) \ 



2;V/x 2 -A 2 C 1 C" CV 2 -A 2 C) ^C 2 ^ 2 -/^)/- 



Now since ?•= tt(/a — \S ft — h*C cos <f>) nearly, it follows that 



r«=2/*r-A s - ^-^-° sin 9 <£. 



After substitutiug this value of r 2 , the equation may be arranged 

 as follows : 



^_ J AV 3A 2 C-V ] 



•;?=JFO ' 1 + 2C 2 ' ft-¥C \ 

 h*u! , sin 2 d> 



+ 2&' ♦V-flO"-^ 



The values of the arbitrary constants A 2 and C, derived in 

 terms of the constants a and e from the apsidal equation (A), are 



C = ^ +2/*'a«(l + e 8 ), A 2 = /i«(l-e 2 )-A 4 (l-e 2 ) 2 . 



If these be substituted in the foregoing equation, the result to 

 the same approximation is, 



V ' 2/j, ' er e 2(i r 



