410 Mr. G. H. Darwin on Variations in the Vertical 



the plane. It is required to find the form assumed by the 

 surface, and generally the condition of internal strain. 



This is clearly equivalent to the problem of finding the dis- 

 tortion of the earth's surface produced by parallel undulations 

 of barometric elevation and depression. It is but a slight 

 objection to the correctness of a rough estimate of the kind 

 required, that barometric disturbances do not actually occur 

 in parallel bands, but rather in circles. And when we con- 

 sider the magnitude of actual terrestrial storms, it is obvious 

 that the curvature of the earth's surface may be safely neg- 

 lected. 



This problem is mathematically identical with that of finding 

 the state of stress produced in the earth by the weight of a 

 series of parallel mountains. The solution of this problem has 

 recently been published in a paper hj me in the ' Philosophical 

 Transactions' (part ii. 1882, pp. 187-230); and the solution 

 there found may be adapted to the present case in a few- 

 lines. 



The problem only involves two dimensions. If the origin 

 be taken in the mean horizontal surface, which equally divides 

 the mountains and valleys, and if the axis of z be horizontal 

 and perpendicular to the mountain-chains, and if the axis of x 

 be drawn vertically downwards, then the equation to the 

 mountains and valleys is supposed to be 



x— —A COS 7, 



so that the wave-length from crest to crest of the mountain- 

 ranges is 27rb. 



The solution may easily be found from the analysis of sec- 

 tion 7 of the paper referred to. It is as follows: — 



Let a, 7 be the displacements at the point x, z vertically 

 downwards and horizontally (« has here the opposite sign to 

 the « of (44)). Let w be the density of the rocks of which the 

 mountains are composed, g gravity, v modulus of rigidity : 

 then 



■=M«f-^1 



1 , rfW 



where ^ 7 .. z 



\V = — gwk e~ x '° cos j- 



From these we have at once 



