Elastic Equilibrium of Semi-Infinite Solid. 



59 



are given at z — o. then we sliall liave tlie following values for the 

 arbitrarv constants: 



B = 



/ + -2// 



i? + 



1 



-In'/.+ ii'l -2 /.+ a)l- 



z, 



D=--^±^/^Z+ 1 



1 



-R, 



C= ^ (Z-B). 



Putting these in (128) and (129) we have the solution corre- 

 sponding to the boundary conditions (130). 



If the traction over the surface is given in the form 



zz = p{r), 

 T>- = T(r), 



(131) 



p and r being any prescriljed functions of r, the corresponding 

 solution can be obtained by making use of the integral theorem 

 (26), on the supposition that the functions p(r) and r(r) do not 

 violate that theorem. 

 Thus 



Î,,. = - /"^(^LrziA-i -i? A- 1+ ^• + '-^^-' . B(k) 



- -— ^ . Z\ A) \e-''J.(kr'dk, 



2(/ + /i)A / 



u,= - f ( -[ZA -i?(A] + -i;t?L.Z(A) 

 ./ \2a ^ 2uU+n^k 



2(/. + /.)A f 



(132) 



and 



rz = f[kz[Z(k)-B k)] + Z[]i)}e-''%(kr)dk, 







I? = ßkz[Z[k)-Bk)] + Bxk)]e'''J,{kr)dk, 

 etc. ; 



(133) 



