of an Elastic Plate under Normal Pressure. 105 



In addition to the above couples there are mean shear 

 stresses ~F 1 and F 2 , on the faces on which P : and P 2 act, these 

 stresses acting in the direction of the ^-axis. The magnitudes 

 of these stresses are 



^-fil*^ • • • • (i9 > 

 F ^=-£|( v ^ (20) 



The directions of these stresses are shown in fig. 4. 



Fig;. 4. 



If the edge of the plate is perpendicular to the #-axis, and 

 if this edge is free, the boundary conditions, according to 

 Kirchhoff, are 



P 1 = 0, s = o/j 

 BQ_. u. „ K • • • ( 21 ) 



21iF 1 



a* 



=0, M 1 =0. 



J 



Poisson thought that F x and Q could be made to vanish 

 separately at the boundary, but KirchhofP showed that the 

 solution of Poisson's differential equation does not contain 

 enough arbitrary functions to satisfy all Poisson's boundary 

 conditions. The reason why Poisson's boundary conditions 

 cannot be satisfied is because his differential equation is 

 derived from assumptions which are not even approximately 



