102 Dr. J. Prescott on the Equations of Equilibrium 



■ Let p denote the external force per unit area perpendicular 

 to the plate. Then Poisson obtains the equation 



f^ll^V 4 ^ (13) 



This equation gives the pressure supported by transverse 

 shearing forces in the plate in the same way that the load on 

 a beam is supported by the transverse shearing force. The 

 stresses which we have denoted by P l5 P 2 , S, are assumed to 

 be zero in arriving at equation (13), but it is the object of 

 this paper to show that these stresses cannot usually be zero. 

 We must now correct this equation by taking into account 

 the effect of the mean stresses P 1? P 2 , and S. 



Let P/ (fig. 2) make a small angle — -tyi with the plane 

 of xy ; and let P/' make an angle tyi' with the same plane. 



Fig. 2. 



Then, since these two stresses act on an area 21idy, the 

 component, in the direction of the 2-axis, of the force due 

 to P x ; and P x " acting on the small rectangle is 



2hdy(-p i "ir 1 " -P^') = ^hdy %%£j& dx. 



But yfr 1 ' = ^~- , and Vi =T > 1 approximately. Therefore 



2A%(P 1 ''t 1 "-P 1 'fi') = 2/^(P 1 |f)^^. 



Thus the component force per unit area on the small 

 rectangle in the direction of the 2-axis due to P x ' and P 3 " is- 



~doc 



O'S) 



Likewise, due to P 2 ' and P 2 " there is a force per unit 

 area in the same direction of amount 



a rMW- 



