Elastic Equilibrium of Semi-Infinite Solid. 33 



X being a real quantity, then 



COshjL' = .9, 



and we have 



cot -^"^ — = — itanh-'' = —ii 



'ho, 2 Vs+1 



sin ^^ = — /sinh.r = — /Cs- — 1)', 



CO, 



sui ~" = /siiilr2a' = •2is{s'- — iy, 



Substituting these values in the equation (7G), we have 



This approximation formula is recommended for the case in which 



For the case ^ > I-, if we put 



(e, - ej' (e, - ej - (e, - ^v))' f 



then we shall have 



Q^v — (o.j) = t', 



and making use of this formula, we shall obtain an approximation 

 formula which is convenient for the case d z>h. 



§26. The expressions obtained in §23 are rather complicated, 

 and the state of deformation can hardly be grasped at a glance. 

 The formula for the displacement at the surface is, however, 

 comparatively simple and can Ije calculated witli any accuracy. 



Putting z = in tlie general formulae, we obtain 



— (r :> a)\ 

 and 



