October i 



1900] 



NATURE 



583 



The roots of this cubic in p are all real if cos'fl>^/8i«V-, 

 :ind the rule of signs shows that there is only one negative root. 

 The curve of no velocity consists then of a ck>sed branch round 

 ihe origin of co-ordinates,the centre, E, of the earth in the present 

 case. Besides this there are two infinite branches which are 

 asymptotic to the parallel lines AB, a'b' represented by 



VH = (• 

 Thus the curve is as roughly represented in Fig. i. The line CD 

 shows the direction of the radius vector from the sun. 



Between the closed curve and the infinite branches v is 

 imaginary, and the satellite must be either within the closed 

 branch or beyond the boundary represented by the infinite 

 branches. The calculation gives very approximately no equa- 

 torial radii of the earth for the greatest distance of any point of 

 the closed branch from the centre. The form of this branch is 

 that of an oval, being slightly longer in the direction towards 

 the sun than in the transverse direction. 



The theorem of Roche which we discuss here is contained in 

 the statement that the atmosphere of a satellite cannot be held 

 together merely by the gravitational attraction of the satellite 

 unless the inequality 



is fulfilled, in which m is now the mass of the satellite, M that 

 of the planet, c the ratio of the square of the angular velocity 



of the satellite's axial rotation to the square of the angular 

 velocity of its orbital revolution, a denotes the satellite's radius, 

 and r the distance of the centre of the satellite from the planet. 

 If the densities of the planet and satellite be 5^,$, and their radii 

 a,,a, and c be unity, that is if the satellite turns always the 

 same face to the primary, we have for the inequality 



■>3. or r>i"44ai 



^-^s. 



It should be mentioned that Roche's investigations embraced 

 much more than this ; they included the determination of the figure 

 of a fluid satellite, and entered into other matters which cannot 

 be discussed here. 



To deal with these questions in an elementary way, 

 consider the important particular case of a spherical swarm of 

 radius a moving round the sun, and turning as a whole about an 

 axis perpendicular to the orbit in the period of revolution, so 

 that it turns the same face always towards the sun. This is, of 

 course, a less general problem than that considered above ; it is 

 indeed the case of that problem in which w is zero It is inter- 

 esting to see from the more general investigation that the con- 

 ition obtained by the consideration of this case is sufficient to | 



give stability for any value of w provided it fulfils the equation 

 of condition stated above. We shall obtain also by the ele- 

 mentary process a wider condition for the case in which w is 

 not zero. This will give the inferior limit assigned by Roche to 

 the distance of a satellite from its primary. 



A particle, of unit mass, say, at the centre, c (Fig. 2), at 

 distance sc ( = r) from the sun, is in relative equilibrium under 

 the sun's attraction and the so-called centrifugal force. That is, 

 we have for that particle 



-_ -n-r = o. 



Again, a particle on the outside of the swarm at the point 

 nearest the sun is at a distance r-a, and under attraction 

 kMI{r-a)-. Hence there is a preponderance of attraction over 

 the acceleration n^r-a) towards s. This excess is 



(r-af 



-«M.-«)=.m( ' -J,.^} 



■ ZkUl 



nearly. This must at least be balanced by the attraction 

 towards the centre, c, exerted by the swarm, if the particle is not 

 to leave the swarm. Hence we must have ^nska^/a- > ;iiMa/r^, 

 or 



as before. The same result would be obtained for a particle at 

 B. In that case the attraction of the sun kM/(r+a)'^ would be 

 insufficient to supply the acceleration n-{r + a) towards the sun. 

 The condition that this should be supplied by the attraction of 

 the swarm is that ^irsr^ should be at least equal to 3M. 



This result holds, of course, for all particles within the swarm 

 on the line sc, for no particle experiences any force on the 

 whole from the spherical layer outside it. 



It is to be observed that a particle at A or B (or on the line 

 sc) is in greater danger of leaving the swarm from the causes 



just explained, than a particle elsewhere on the spherical 

 surface. 



If the particles of the swarm have other angular velocities than 

 that supposed above about an axis through its centre perpen- 

 dicular to the plane of the orbit, the investigation will run as 

 follows. Suppose applied to each of the particles a force per 

 unit mass equal and opposite to that, «V, exerted by the sun on 

 the central particle. This will have no effect on the relative 

 motions of the particles or on the figure of the swarm. Upon 

 the particle nearest the sun the force per unit mass toward the 

 sun is now 



(r-af r3 ' 



if wi be the angular velocity t'n space of the radius vector drawn 

 from the centre to the particle (that is, not the relative angular 

 velocity a> above, but « f «)• This must at least be balanced by 

 the attraction towards the centre exerted by the swarm if the 

 particle is not to leave it. Thus we have ^irisa > 2i'Ma/r^ + (o^a, 

 or since ^-M = w^r* 



(-50 



M. 



Thus if the swarm as a whole make one rotation in the period 

 of revolution round the sun, u>j7«'=i> and we obtain the same 

 result as before. 



Let now the swarm of particles be replaced by a spherical 

 planet with an atmosphere composed of discrete small particles, 

 the whole being held together by gravitational attraction alone. 

 Then if the mass of the planet be denoted by in, the inequality 

 ifitsr^ > 3M becomes ^wsa^ > ^Ma^/r^, that is w//M > 30'/^. This 

 is to be fulfilled if the atmosphere is not to be dissipated by tidal 



NO. 16 I 5, VOL. 62] 



