52 The Rev. A. Thacker on a case of disturbed Elliptic Motion. 



Since the force is central, there cannot be more than two ap sidal 



distances. Let them be a and /3, the former being the greater. For 



dr 

 each of these values of r, -j must be zero. We have, therefore, 



= - Ca 2 - h°- + 2/i« - (m'u 4 , 

 0=-C/3 2 -/i 2 + 2}i/3-im'/3 4 ; 



whence 





If these values be substituted for C and A 2 , it will be found that 

 ■W-k*+2pr- ^r 4 =(u-r)(r- £)j-^ +/t '(r + «)(>- + /3) j, 



whence 

 dt 



dr=± 



V /( a -,-)(.'-/3){^+, J '(,- + «)(,.+/ J )j 



the itpper or lower sign being taken according as the body is 

 moving from or towards the nearer apse. If the expression 



■< — 1—£ + //,'(;• + «) [r + /S) f- be expanded in a series ascending 



by powers of fj,', the value of t in terms of r may be found by 

 tegration to any required degree of accuracy. Neglecting 



powers of /J above the first, putting k for — • — = — > anu U for 



(« — ;•)(?•— /3), we have 



V- 



2fM dt r 1 ?- 3 +(a + /3)r 2 + «/3r 

 — ^ k 



« + /3 dr a/\J 2 -/U 



Integrating, and supposing t to begin when the body is at the 

 nearer apse, we obtain 



/I^T , 1, , m ,a + /3-2r _, 



-~ k( a + /3)(ll^ + 10^/3+11^)^^^-^— 



+ ^t{8r 2 + 22( a + / S)r + 33« 2 + 74«/3 + 33/3-} ^\j 4 



If we make cos" 1 ^ - = u, it follows that 



a — p 



»-=o(*+/3)-o(*-i9)cos«j ... (A) 



