212 SEE— RESULTS OF RECENT [April 23, 



satellites in like manner are within the closed surfaces about their 

 several planets ; and Dr. Hill remarks that this arrangement is 

 necessary to secure stability. If a satellite is once within this region, 

 with the surface of zero velocity closed about it, it cannot escape, 

 but will always remain attached to the planet, and its radius vector 

 will have a superior limit. How the moon and other satellites came 

 within these closed regions Dr. Hill did not inquire ; and subsequent 

 investigators appear to have supposed that as these bodies cannot 

 now escape from their planets, so also they cannot have come in from 

 a remote distance, but must have originated where they now are. 

 This is the view put forth by Moulton in his discussion of Professor 

 W. H. Pickering's suggestion that Phoebe had been captured by 

 Saturn ; but such reasoning is easily shown to be erroneous by the 

 following considerations : 



(3) Jacobi's integral, as originally given by him, is based on the 

 differential equations for unrestricted motion in empty space, and 

 no account is taken of the additional terms which must be added to 

 the differential equations of the motion of the sun, planet and 

 particle, when the motion is very slightly conditioned by the intro- 

 duction of a nebular resisting medium, such as existed in the early 

 history of our system, and is now observed to be widely diffused 

 throughout nature. Jacobi's original integral, therefore, requires 

 the addition of a secular term to represent the actual movement of a 

 sun, planet and particle ; and the complete expression for any 

 particle whose coordinates are xi, yi, Zi becomes 



,2 , 2(1 -/i) , 2// 



» ' -^ t ' _ // .. .. \2 ■ .. 2 , .. 2 ' 



-=C,-\-a^t. (i) 



The secular term atti makes the constant d increase with the time. 

 («) Now the surfaces of zero relative velocity, which define the 

 closed spaces about the planets, have larger values of C the nearer 

 we approach to the sun or planet. This is easily seen in the accom- 

 panying plate from Darwin's celebrated memoir on " Periodic 

 Orbits." When the particle or satellite revolves against resistance, 

 therefore, the second member of (i) increases, and there is a secu- 



