of an Elastic Plate under Normal Pressure. 103 



Again, let — (/>/ denote the small angle which the force 

 due to S/ makes with the xy plane, and ${' the corresponding 

 angle for Si". Then the component force parallel to the 

 ~-axis due to these two forces is 



, da 

 But^'=|^, and S/^S^S. Therefore . 



oy u . 



2hdy($ 1 ''cf> 1 ,f -$ 1 ><p 1 >) = 2h^(^)dxdy. 



ox\ oy J 



Likewise, the corresponding force due to S 2 ' and S 2 " is 



Therefore the total force per unit area in the direction of 

 the £-axis due to all the mean stresses acting on the edges 

 of the element dx X dy is 



which, by means of equations (10) and (11), simplifies to 



l^^' 2 cty 2 d?/ 2 d^ 2 *dxo~y ' "daisy) ' 

 This is the expression that must be added tojt? in equation 

 (13), since, in that equation, p is the force per unit area in 

 the direction of the ->axis which has to be supported by the 

 transverse shear stresses. Thus the corrected form of 

 Poissom's equation is 



r 61 _ a! vw iu, J a^.^ + ^ a- i 



p being, in this equation, as in Poissoir's equation, the 

 external force per unit area. 



This last equation takes the place of Poisson's equation. 

 Since it contains two unknown functions, w and <£, it must 

 be combined with equation (12), which we rewrite here, 



The residual stresses parallel to the middle surface after 

 the mean stresses P 1? P 2 , S, are taken away, are proportional 



