58 THE EQUILIBRIUM OF ELASTIC 8OUDa 



Let A, be the pressure on one side of the plate, and A, that on the other. 



Let p and q be the pressures in the plane of the plate at any point, p 

 acting in the direction of a tangent to the section of the plate by a plane 

 passing through the axis, and q acting in the direction perpendicular to that 

 plane. 



By equating the forces which act on any particle in a direction parallel to 

 the axis, we find 



dr dx , dp dx d*x . . dr 



By making p = Q when r = in this equation, when integrated, 



The forces perpendicular to the axis are 



fdr\ dpdr d*r n ,. dx 



Substituting for p its value, the equation gives 



(A.-&.) (dr dr dx\ 

 1 t \dsdx + ds)' i 



The equations of elasticity become 



'dr ds d*x ds d*r\ ,_ v 



r-a- j.' A.- d& r d fi r \ j XT.- 



Uinerentiatmg -^ = -5- ( rl, and in this case 



d8r _ dr dr 



dr ds ds ds ' 



By a comparison of these values of , 



.. _ P +L?-2 + ^4,-/ ' -1^* + 



9p 3m/\ P 2 ) + m + dsm* r <) 3m/ dr + dr 



ds 



