JOURNAL OF THE COLLEGE OF SCIE^'CE, IMPERIAL UNIVERSITY, TOKYO. 



VOL. XXXVII., ARTICLE 7. 



On the Elastic Equilibrium of a Semi=Infinite Solid 



under given Boundary Conditions, with 



some Applications. 



By 

 Kwan-ichi TERAZAWA, JHnakushi, 



I. Introduction. 



§1. The statical problem concerning an infinite elastic solid 

 bounded by a plane subjected to a given distributiou of traction or 

 deformation has attracted the attention of numerous eminent 

 elasticians. The first solution for the case of a purely normal load 

 was given by Lame axd Clapeyron^^ by means of Fourier's 

 theorem, through which an assigned function of two variables is 

 expressed as a quadruple integral. The credit of first improvement 

 on this subject may well be claimed by J. Boussinesq,'^ who 

 introduced several kinds of potentials — direct, inverse and logari- 

 thmic with three variables — into the theory of elasticity, and 

 opened a new field of treatment in it. Almost all conceivable 

 cases have been solved by him, especially in relation to what takes 

 place at the boundary surface. Besides Boussinesq, many other 

 authors have touched on this problem, employing the method of 

 integration by Green's functions. Not long ago, Prof. H. Lamp.^^ 

 solved a special case of this problem, viz. that in which the 

 boundary condition is a normal pressure symmetrically distributed 

 about a point on the surface, by making use of the integral theorem 

 of Fourier's type concerning Bessel function of the zeroth order; 

 and thus. Lamé and Clapeyron's method, which was considered to 



1) Crelle's Journal, vol. 7 (1831) p.p. 400-40i. 



2) Application des Potentiels, Paris, (1885;. 



3) Loncl. Math. Soc. Proc. vol. 34 (1902) p. 276. 



