272 The Philosophical Society of Glasgow. 



On the Variation ofthePeAoclic Times of the Earth and inferior Planets, 

 produced by flatter Falling into the Sun. By PfiorESSOE Wm. 

 Thomsok. 



It may be remarked, in the first place, that the absolute effect on 

 the periodic time of Mercury, producible by such a distribution of 

 planetary matter as M. Le Verrier concludes must circulate between 

 Mercury and the Sun, is not discoverable. The true mean distance of 

 Mercuiy from the Sun is not, in fact, known with sufficient accuracy 

 to allow us to judge whether or not the central force correspond- 

 ing to its periodic time, when compared with the forces experienced by 

 the other planets, deviates from the law of inverse squares of distances 

 from the Sun's centre to such an extent as must be the case if M. Le 

 Verrier's disturbing planetary matter is altogether inside Mercury's 

 orbit. But it does not follow that the periodic time of Mercury, or 

 even that of Venus, and the Earth may not be sensibly influenced by 

 the falling in of portions of that matter to the Sun. To discover 

 the general character of this influence, and to estimate its amount, we 

 may first consider the resultant force experienced by a planet under the 

 joint influence of the Sun and a concentric cii-cular ring in the plane of 

 its orbit. It is easily seen that the Sun's force must be diminished by 

 the attraction of the ring, if this lies outside the planet, but, on the 

 contrary, increased, if inside. If the radius of the ring be very small 

 in compai'ison with the distance of the planet, it is clear that to a first 

 approximation the attraction of the ring may be calculated, by sup- 

 posing its mass collected at its centre. The full expression for the 

 resultant attraction of the ring being the following convergent series : — 



where m denotes the mass of the ring, r its radius, and a the distance 

 of the planet from its centre, shows that the resultant force is in reality 

 somewhat greater than it would be if the mass were collected at the 

 centre. The planet's orbit being nearly circular, the attraction of such 

 a ring as we have supposed will be represented to a second ap- 

 proximation bj-^ supposing a mass greater than its own in the ratio of 



3 /r\2 

 ltoI+x( — ) > collected at its centre, or which would produce the 



same effect, distributed uniformly over the Sun's surface. Hence the 

 gradual falling in of such a ring to the Sun will diminish the force 



