1880.] History of Planet and Single Satellite. 



2Ho 



From this equation we get the curious relationship 



B x* 



Ai ' (x±-hx*+iy 



(20). 



This last result will obviously be equally true even if all the roots 

 of x* — hx 3 + 1 = are imaginary. 



In the present case the complete solution of the problem is com- 

 prised in the following equations : — 



, ■ * r hp x-oT\ 



(x oo a)8a x exp. |_4( ai 2 + i8 2) arc tan ~p~ J 



■ ; A — 



(ajcob)^^— «)2 + 02] 8(ai2+i81 



A* (sjj*— AajS+l) 

 n—li—x. 



. (21). 



It is obvious that the system can never degrade in such a way 

 that as should pass through one of the roots of the biquadratic 

 x 4 — Aas 3 + 1=0. Hence the solution is divided into three fields, viz., 

 (i) 35= -|- qo to a?=a; here we must write x — a, x — b for the a;coa, 

 a? cob in the above solution; (ii) aj=a to aj=b'; here we must write 

 a — x, x~b (this is the part which has most interest in application to 

 actual planets and satellites); (iii) x=h to x— — oo ; here we must 

 write a — x, b — as. When x is negative the physical meaning is that 

 the revolution of the satellite is adverse to the planet's rotation. 



By referring to (4) and (6), we see that i must be a maximum or 

 minimum when « = 2Q, and e a maximum or minimum when n==^Q. 

 Hence the corresponding values of x are the roots of the equations 

 a? 4 — foe 3 + 2=0, and a? 4 — hx s + j-j- =0 respectively. 



Since 



— a. Y {x — at) +p~ 



x*-hx 3 + l ^as-a ^as-b 2(^ + £ 2 ) [>-«) 3 + /3 3 ] 

 Therefore 



+ or [ - *i O - a) + /3 3 ] (x - a) (x - b) . 



2( ai 2 + /^) 



