The question of the stability of the motion under these condi tions has engaged the attention of at least three mathematicians. The first in the field was L. J. Linders, who followed a method in dicated by Poincare (Lecons de Mec. Cel §64), and obtained a first approximation to the resulting motion, which he found to be stable within certain limits of departure from the Lagrangian configuration. He found that the period of revolution of a Trojan planet would oscillate between 4,222 and 4,442 days, that of Jupi ter being 4,332 days, or 11.86 years. The greatest departure in angle from the mean position is I7-1-°. He found that there would be long period oscillations whose duration is about 15o years. Prof. E. W. Brown in 1911 studied the problem on the same lines as Prof. Sir G. Darwin had used in his investigations on periodic orbits. He treated the motion of Jupiter round the sun as exactly circular, and referred the motion of the asteroid to the revolving line joining Jupiter to the sun. The nearest approach of the aster oid to Jupiter compatible with stability was found to be about two astronomical units, or 231°. W. M. Smart (1917) carried Lind ers's work to a further order of approximation, but he still consid ered Jupiter as moving in a circle. He showed that the oscillations of the Trojans could be divided into two classes: (r) those arising directly from the inclination and eccentricity of the Trojans; the period of these is about 12 years. (2) a much slower oscillation, which may very approximately be represented as harmonic motion in an ellipse whose shorter axis is directed towards the sun, the longer axis being 184 times the length of the shorter. The period in which this ellipse is described is 147.82 years, just 121 times Jupiter's period of revolution.
Two further papers by Prof. E. W. Brown deal more directly with the actual Trojans than his earlier one, which deals with ideal cases. He reduced the theory to a numerical form, and showed his approximations suffice to represent the observations of Achilles, extending over many years, within a few seconds of arc. He
also examined the stability of the motion; it is stable as far as the sun and Jupiter are concerned, but the action of other planets es pecially Saturn may in a very long period (of the order of a million years) cause the Trojans to approach too near to Jupiter, after which the character of their motion will be changed, and they will cease to be Trojans. However, since the age of the planetary sys tem is estimated as thousands of millions of years, and at least six Trojans still survive, it would seem that such disturbing action on Saturn's part can only under exceptional circumstances lead to the expulsion of a Trojan; otherwise none of them would survive, for it does not appear likely that Saturn can reverse its action, and turn non-Trojans into Trojans. Any asteroids that had once been Trojans would continue to make near approaches to Jupiter's orbit. It seems possible that the remarkable asteroid 944 Hidalgo (see MINOR PLANET) may once have been a Trojan, but no other aster oids are known that fulfil the conditions.
The existence of the Trojans suggests that in the early days of the planetary system a large amount of scattered matter was cir culating round the sun at the same distance as Jupiter, of which the Trojans have survived in virtue of the stability of their con figuration, while the remainder has had its period changed or been absorbed by Jupiter.
Some may imagine that, as there are several Trojans on the same side of Jupiter with a common point as their centre of oscil lation, there is some danger of collision between them. Such dan ger is very remote, for the length of their swings exceeds roo,000, 000m., and these are performed in different directions; also the times at which they reach the ends of their swings are different, though the periods of the different swings are nearly the same for all of them.