394 



NATURE 



[February 24, 1898 



invisible in the sound part. An analogous process may 

 .also be seen in the grand caiion of the Yellowstone 

 River, where the rhyolites have been weathered into a 

 white or yellowish, mealy-textured rock along the great 

 fissure, which allowed a freer passage to, and still emits, 

 in diminished quantities at the bottom of the gorge, 

 Tieated water and steam from the dying-out volcanic 

 furnace below. 



So along the Armenian lava-flow water easily got in 

 •from the surface, or followed the longitudinal cracks in 

 the rapidly cooled rock, and, being itself heated by the 

 process, acted upon the silicates, and carried off the more 

 soluble portions. A diminution in volume accompanied 

 'the process, and the reduced rock broke up. 



May be the old valley down which this lava flowed 

 was an ancient outfall of Lake Gokcha before its rim 

 liad been built up as high as it now stands. Perhaps 

 at one time some of the water of the lake found its way 

 through fissured masses to lower levels, and carried on 

 the work of destruction within the rock itself 



We want many more facts respecting this most in- 

 teresting district— we would like to know the depth of 

 the lake, the direction of any observed lines of fissure, 

 a.nd if there is any evidence of the waters of the lake 

 having ever been suddenly drained off into the valley of 

 the Araxes. We hope, therefore, that Mr. Loewinson 

 Lessing, whose knowledge and skilful arrangements 

 enabled us to see so much of the district, and whose 

 courtesy and thoughtfulness made our excursion so 

 pleasant, may be able to carry on the work for which 

 he is so well qualified, and will communicate to the 

 world the result of his further investigations. 



T. McKenny Hughes. 



PERIODIC ORBITS. 



T7OR some years past. Prof G. H. Darwin has been 

 -■- engaged on the numerical solution of a particular 

 case of the problem of three bodies, and at different 

 times he has given some account of the progress he has 

 made. He has now collected his very extensive material, 

 relating to both the mathematical methods employed 

 and the discussion of the numerical results, into one 

 compact summary under the title of " Periodic Orbits," 

 which appears in Acta Mathematica, vol. xxi. The 

 special case treated by Prof. Darwin refers to one of 

 three classes into which M. Poincare has divided the 

 periodic solutions of this problem. In this class, the 

 motion is entirely in two dimensions, and the excentricity 

 of the planet's orbit is very small ; but Prof Darwin 

 further supposes the perturbed body to have infinitely 

 small mass, and the planet's orbit to be absolutely 

 circular. The discussion of even this one class has had 

 to be restricted in the course of the work, on account of 

 the heavy arithmetical labour which the method of 

 tracing the orbits by mechanical quadratures involved. 

 Retrograde orbits have not been considered, and the 

 motion of superior planets is still engaging Prof Darwin's 

 attention. Some thirty examples of periodic orbits have, 

 however, been examined ; and though the author may 

 speak of his results very modestly, there is no doubt but 

 that his conclusions will be welcomed as a most interesting 

 •contribution to the study of celestial mechanics. 



Prof. Darwin defines a periodic orbit as one in which 

 a third body can continually revolve so as always to 

 present the same character relative to the two other 

 bodies of the system. These orbits are not necessarily 

 confined to a single revolution round the primaries, or 

 round any other point in space, but the difficulty of the 

 determination of the path increases with the number of 

 circuits described, and on that ground the present 

 treatise is confined to the examination of " simple 

 periodic orbits," or those which are re-entrant after a 



NO. 1478, VOL. 57] 



single circuit, though loops may, and do, occur in the 

 orbit. In the system considered, the distance between 

 the sun and the principal planet, here called Jove, is 

 taken as unity, and the ratio of the mass of the sun to 

 that of the planet as lo : i. This hypothesis differs 

 considerably from the actual circumstances prevailing 

 in our system, but it offers the advantage of exaggerating 

 all the phenomena of perturbation, and permits the clear 

 exhibition of the deductions in diagrams that are easily 

 appreciated. Some of the stellar systems may offer 

 conditions more nearly parallel to those here assumed. 

 A looped orbit has been suggested in the case of one of 

 the components of f Cancri, though possibly with in- 

 sufficient data, and in some other cases, the presence of 

 a disturbing body seems likely to produce an orbit of 

 very irregular form. 



We have accustomed ourselves to consider the relations 

 of superior and inferior planets and of satellites to be 

 fixed and definite, but Prof Darwin traces the conditions 

 under which these forms cease to be permanent, and 

 when consequently the third body of a system can 

 assume the characteristic motion of either an inferior 

 planet or a satellite. With a particular value for the 

 constant of relative energy, it is possible for both kinds 

 of planetary and satellite motion to become confused, 

 and for a body which originally started in one way to 

 exhibit the peculiar motion of either of the other two. 

 Prof Darwin began his numerical work by an assumed 

 case in which it was possible for an inferior planet and 

 satellite to interchange their parts. The satellite was 

 made to start at right angles to a line joining the sun 

 and Jove, at a distance of ro8 from the sun, Jove's 

 distance being unity. The resulting orbit is fully drawn 

 and shows how the body hangs in the balance, between 

 the two centres, before the elliptic form of the orbit 

 asserts itself, as the body approaches the sun. Starting 

 the satellite from a conjunction remote from the sun, but 

 at slightly different distances from Jove, it is found that 

 the resulting orbits show a great diversity of character, 

 which cannot always be foreseen. Perhaps the most 

 remarkable curve in this family arises when the satellite 

 starts at a point 1*095 ^"^om the sun. After making a 

 loop, the satellite recrosses the fine of conjunction and 

 moves directly towards the planet, so that it is impossible 

 to determine its subsequent path without very accurate 

 computation. " I do not think," says Prof. Darwin, 

 " any one could have conjectured how the body should 

 have been projected so as to fall into Jove." The positions 

 which give rise to periodic orbits are shown by the 

 distances from the sun at which the curve meets the line 

 of conjunction after one complete circuit. If for two 

 selected points of projection, the curve returns to this 

 line at places alternately nearer to and more remote 

 from the sun than those from which it originally started, 

 then there must be some point intermediate between 

 these selected points at which the curve will be re- 

 entrant. Other forms of orbits giving rise to distinct 

 families, have been computed, and drawn, when the 

 satellite is projected from points intermediate between 

 the sun and Jove, and also from conjunction on the side 

 remote from Jove. Most of these orbits do not possess 

 the character of stability, a point which the author has 

 considered with as much care as the form of the orbit 

 itself It has been questioned whether all orbits are not 

 essentially unstable, if the number of revolutions be 

 sufficiently great. The result of the present investiga- 

 tion is to show that orbits may be stable if the per- 

 turbation of Jove by the planet can be neglected. This 

 is the only approximation that Prof. Darwin has per- 

 mitted himself, and he remarks " that for a very small 

 planet the instability must accordingly be a very slow 

 process, and I cannot but beheve that the whole history 

 of a planetary system may be comprised in .the interval 

 required for the instability to render itself manifest." 



