394 Mr. Davison on the Relation between Size of a Planet 



throw the whole contraction into the vertical dimension, every 

 shell will descend too far. It is obvious therefore that, on 

 both accounts, we get too much compression or folding. 

 These are the suppositions that I made to simplify the cal- 

 culation. The amount of sinking of the shells, and consequent 

 compression owing to loss of room, is then simply caused by 

 the total contraction in the vertical of the matter interior to 

 each shell ; and does not depend upon when the contraction 

 occurred, or how long it took ; and is therefore independent 

 of the time ; and is so far not according to nature. But 

 nevertheless, seeing that it gives a superior limit to the com- 

 pression and consequent elevations, if we find the result too 

 small to suit the observed facts, it furnishes an a fortiori 

 argument against the so-called " contraction theory." 



The tidal theories of Professors Pierce and Darwin, ap- 

 pealed to in section 19 of Mr. Davison's paper to supplement 

 the contraction-theory, involve considerations of so much 

 complexity that I make no reference to them here. 



LI. Note on the Relation between the Size of a Planet and the 

 Rate of Mountain-building on its Surface. By Charles 

 Davison, M.A., Mathematical Master at King Edward's 

 High School, Birmingham* 



1. TNa recent paper f I have investigated the distribution 

 J- of strain in the earth's crust resulting from secular 

 cooling, supposing the earth to have been initially at a high 

 temperature and practically solid throughout. Other condi- 

 tions being the same, it is not difficult to show that, the 

 smaller a planet, the more rapid is the rate of mountain- 

 building on its surface, at any rate in the early periods of its 

 history. 



Supposing the planet to consist of an uncooled spherical 

 nucleus surrounded by a series of very thin concentric sphe- 

 rical shells, of internal radii ?* , rj , . . . . respectively (begin- 

 ning from the nucleus), then the change of radius of the inner 

 surface of the (n+l)th shell in a given time is proportional 



n 



where Sd is the difference in the rates of cooling of two con- 

 secutive shells, and e is the coefficient of expansion J. If this 



* Communicated by the Author. 



t " On the Distribution of Strain in the Earth's Crust resulting from 

 Secular Cooling, &c," Phil. Trans, vol. 178 (1887), A. pp. 231-242. 

 X Ibid. p. 233. 



