AND DIAGRAMS OF FORCES. 181 



however, a general solution of these equations, as may be easily seen by taking 

 the case in which p^ and p yz are both zero at all points. In this case, since 

 there is no tangential action in planes parallel to xy, the stresses p m> p,^ and 

 p m in each stratum must separately fulfil the conditions of equilibrium, 



d d d d 



The complete solution of these equations is, as we have seen, 



_df df _d*f 



~ Pxy ~ dxdy' Pm ~ 



where f is any function of x and y, the form of which may be different for 

 every different value of z, so that we may regard f as a perfectly general 

 function of x, y, and z. 



Again, if we consider a cylindrical portion of the body with its generating 

 lines parallel to z, we shall see that there is no tangential action parallel to z 

 between this cylinder and the rest of the body. Hence the longitudinal stress 

 in this cylinder must be constant throughout its length, and is independent of 

 the stress in any other part of the body. 



Hence p a = <$>(x,y), 



where <f> is a function of x and y only, but may be any such function. But 

 expressing the stresses in terms of F under the conditions p xz = 0, p yi = Q, we 

 find that if F is a perfectly general function of x and y 



, 



-j r = 0, and -T y- = , 

 dxdz dydz 



d F dF 



whence it follows that -j and -7 are functions of x and y only, and that 



dF 



-7- is a function of z only. Hence 



F=G + Z, 



when G is a function of x and y only, and Z a function of z only, and the 

 components of stress are 



<PG<PZ d*Gd?Z (d'GtfG ( 



~ - ~ 



