570 PLANET present is concerned. Now take a point at distance d from the centre of the planet, and on the line joining the centres of the sun and planet; then the attractions exerted by the planet and the sun on a particle placed at this point are respectively proportional to and ^-5-; an( l * n order that these may be equal we must manifestly have -5^=7^5 or d= ~ .D. Again, if d' be the distance of T M+ Vm> a point on the line joining the centres, but be- yond the planet, then clearly we must have, for the attractions to be equal, the relation f =-ri7 -D. Hence or, since m is very small compared with M, d+d' =2 v5-D, very nearly. This is in reality the JVI diameter of the spherical domain of the planet ; for it is easily shown by a geometrical con- struction that the surface where the sun's in- fluence and the planet's are exactly equal is a sphere. Thus let P be the planet, S the sun, Q a point where the planet's influence and the sun's are equal. Then - p - = -, or PQ : SQ :: VM : ym, a constant ratio. But QA, the bisector of the angle PQS, divides PS so that PA : AS : : PQ : QS ; hence A is a point where the planet's attraction equals the sun's; and QB, the bisector of the angle PQF, divides PS produced so that BP : BS :: PQ : QS. There- fore B is another point where the planet's at- traction equals the sun's. And the angle BQA is a right angle, because its parts BQP and AQP are respectively the halves of the angles FQP and SQP, which together make up two right angles. Hence Q is a point on a circle with BA as diameter, and is therefore a point on a spherical surface having BA as diameter. Also manifestly PB is the d', and PA is the d, of the above demonstration. Accordingly the radius of the planet's spherical domain is = V-ji-'Dj or varies as the square root of the planet's mass and the distance of the planet from the sun, jointly. It is easy therefore to calculate the value of this radius for the sev- eral planets. The following table presents the results as calculated by the writer, where the column of velocities is added to illustrate a relation referred to above : PLANETS. Radius of spherical domain. Velocity of bodies at- tracted to the sun from great distances when at distance of these planets. 16,077m. 41 '4 m. per second. 110,850 " 30-3 The earth and moon 163,880 " 85,265 " 25-9 20-8 Jupiter 14,701,000 " 11-3 14,717,000 " 8-3 " 11,114,000 " 5'9 " Neptune. 20,038,000 " 4-7 It will be noticed that the domain ascribed to the earth and moon does not really extend so far as the moon, and in making the calculation the mass of the moon should perhaps not have been added. It is easy to make the necessary correction; for the mass of earth and moon being 1,012 where earth's mass = 1,000, the square root of the mass is greater in the ratio 1,006 : 1,000, or by less than the 160th part ; 162,880 represents the radius thus reduced with sufficient approximation. The mass of the moon being about the 81st part of the earth's, the moon's attraction sphere, estimated solely with reference to the sun, has a radius equal to one ninth of the earth's, or to about 18,100 m. The case is quite different with Jupiter's satellites and Saturn's, seeing that the outermost satellite of Jupiter travels at a dis- tance of only 1,150,000 m. from the planet's centre, and the outermost of Saturn's at a dis- tance of 2,368,000 m. from Saturn's centre, and both these distances are far less than the radius of the spherical domain of either plan- et. The near equality of these domains, not- withstanding the great disparity between the two planets as to their mass, illustrates the effect of distance in diminishing the energy of the sun's action. This is still more strikingly shown in the case of Neptune, whose domain enormously exceeds that of any other planet. When we consider further that matter which has been drawn sunward from interstellar space travels with a velocity proportioned to the square root of the distance from the sun, and therefore moves much more slowly at Neptune's distance than at that of any other planet, we perceive that our estimate of Nep- tune's relative power must be still further in- creased ; and perhaps in such considerations as these we may find an explanation of the anomalous position of Neptune in several re- spects. "We have seen that Bode's law is nc fulfilled in his case. Moreover, the largenc of Neptune's mass compared with that ol Uranus is remarkable. A certain law coi have been recognized in the arrangement ol the masses had the mass of Neptune been 1( than that of Uranus. For in the inner farnil: of planets we perceive that the masses inert with distance from the sun to a maximui and then diminish, an explicable arrangement : and in the outer family we should have had in the case supposed a continuous decrease out ward ; but the decrease down to a minimum f ol
