﻿in Small Orbits. 511 



which they took no part in producing; but if surrounding a 

 star equal in mass to our sun, an envelope of hydrogen tra- 

 versed by meteors would acquire from pressure alone a tempe- 

 rature many thousand times as great as that arising from its 

 combination with oxygen. The disproportion in the efficiency 

 of the two causes will appear still greater when the slowness with 

 which large collections of different gases can mix, so as to afford 

 scope for the play of affinity, is contrasted with the rapidity with 

 which a host of meteors in the supposed case would make their 

 terrific influence felt throughout an extensive region of a solar 

 or a planetary atmosphere. 



In investigating their form and stability in small orbits, I shall 

 regard the satellites as fluid ; for when their size is considerable 

 the resistance of solid matter cannot occasion any wide depar- 

 ture from a figure of equilibrium. The mechanical theory of the 

 resistance of solids shows that passive strength increases in a 

 less rapid ratio than weight, and that large structures are weaker 

 in proportion to their size than smaller ones. But in bodies as 

 large as worlds the restraint from solidity is far less than that 

 given by theoretical estimates, as a strain extending over many 

 hundred miles must fall on some localities in an undue propor- 

 portion ; and in the planetary structure rents would be gradually 

 produced, its several parts yielding in succession to a force which, 

 if duly distributed as theory supposes, might be successfully 

 resisted by the solid mass. Accordingly, in questions relating 

 to the form of a large satellite, the gravity at its surface and 

 the radius of the smallest circular orbit which it can describe, 

 no serious error can arise from adopting the hypothesis of flui- 

 dity. A source of greater discrepancy in the results for differ- 

 ent bodies would be found in the heterogeneous character of 

 their materials, and in the variable rate at which density might 

 decrease from the centre of each to its surface. But the maxi- 

 mum extent to which this cause can affect the result of our 

 inquiries may be determined by selecting two extreme cases, in 

 one of which the satellite is regarded as homogeneous, while in 

 the other the matter at the centre is supposed to be infinitely 

 dense in comparison with that by which it is surrounded. A 

 similar course has been adopted by writers on the figure of the 

 earth. 



I shall take up first the case in which the central matter is so 

 exceedingly dense that it may be supposed to monopolize the 

 whole of the attractive energy, and accordingly that gravity at 

 every part of the surface of the body is directed to the centre. 

 Let D be the distance between the centres of the primary and of 

 the satellite, M and m the measures of the attractions of the two 

 bodies at the distance k, and x, y, and z the rectangular coordi- 



