176 Dr. C. Chree on Applications of Physics 



the direction of " gravity " at the surface. Consider, for 

 example, the influence of the ordinary ocean tide at a point 

 inland near the shore. At high tide there is on the sea- 

 bottom a pressure exceeding the mean by an amount corre- 

 sponding to the height of the water above its mean level ; 

 this will tend to make a naturally horizontal plane dip towards 

 the sea. At the same time the surplus volume of water will 

 give a horizontal component to what we may regard as 

 normal " gravity " in the neighbourhood. This second effect 

 has been called attention to, in this very case, in Thomson 

 and Tait's ' Natural Philosophy/ art. 818, where will be 

 found a numerical estimate for a specified set of conditions. 

 What a spirit-level shows is the plane perpendicular to 

 gravity — including " centrifugal force " and all disturbing 

 forces. We are thus obliged to consider both effects before 

 attempting numerical estimates. 



Direct Pressure Effect, Fundamental Formula?. 



§ 5. In the following calculations the earth is treated as an 

 isotropic elastic solid, principal weight being attached to the 

 results obtained by supposing the material incompressible. 

 Also, as we are primarily interested in the consequences of 

 pressure applied over limited areas, the loaded surface is 

 treated as a horizontal plane, on the lower side of which the 

 material extends to infinity. On these hypotheses we are 

 enabled to make use of the very interesting and important 

 results established by Professors Cerruti and Boussinesq. 



A convenient English abstract * of Boussinesq's work is 

 contained in Todhunter and Pearson's ' History of Elasticity,' 

 vol. ii. part 2, arts. 1492 et seq., from which the following 

 formulae are quoted, the only variation being the use of 

 Thomson and Tait's notation for the elastic constants. 



The origin of coordinates lies in the undisturbed surface, 

 taken as the plane of ay, the positive direction of the z axis 

 being downwards into the earth. The normal pressure applied 

 to the element dw of surface is p da), where p is supposed of 

 course a known function of a, y. 



u, v, w denote the components of elastic displacement, 

 n the rigidity, and rj Poisson's ratio for the material. 



The displacements at any point x } y, z in the material are 

 as follows : 



* Chapter ix. vol. i. of Love's ' Treatise on. . , .Elasticity ' may also 

 be usefully consulted. 



