1880.] History of Planet and Single Satellite. 259 



direction and magnitude. Hence if we draw a parallelogram of 

 which the diagonal is h (the resultant m. of m. of the system), and of 

 which the sides are n and x, inclined respectively to the diagonal at 

 the angles i and j, we see at once that 



smj n 



If i and j be treated as small this reduces to (7). 



Again the consideration of this parallelogram shows that 



}fi — n? -(- a? 2 + 2nx cos (?' +j) , 



which expresses the constancy of moment of momentum. If the 

 squares and higher powers of be neglected, this becomes 



h=n+x (8). 



Equation (8) may also be obtained by observing that dn/dt + dx/dt—0, 

 and therefore on integration n + x is constant. It is obvious from the 

 principle of m. of m. that the planet's equator and the plane of the 

 satellite's orbit have a common node on the invariable plane of the 

 system. 



If we divide equations (4) and (6) by (3), we have the following- 

 results : — 



'f=J(.-!) ri 7 ov. 



% ax zn \ % / n — Q 



lde_ 1 11m-18Q 



e dx 2x n — Q 



But from (7) and (8) 



l XX 



also Q=x~ s , and n — li — x. 



Hence (9) and (10) may be written 



A loo- i= - 1 x*(h-x)- 2 \ 

 dx & 2 x(Ji—x) x z (h—x) — l I 



d, 1 Ux s (h-x) -18 ( 



— loge= — . -i 



dx & 2x x*(h-x)-l J 



(10). 



(11). 



^Now 



2x ( h - x) { x* {li - x) - 1 } x(li-x) 2x±- hx* + 1 



Therefore * fagfeLl- * +* f .... (12). 



dx x li — x 2x 4: — nx 6 + l 



Also 



lW(h-x)-18 _9 7 x*(fB—K) 



2x{x s (h-x)-l} x 2 x^-hxZ + l 



