18 MR CLERK MAXWELL ON 



If -T- = z, F becomes Airy's function of stress in two dimensions, and we have 



d 2 Y d 2 F d 2 F 



Pxx ~ P dy 2 ' Pvv - 13 cW ' P *» - P d^dy • 



The system of stress in three dimensions deduced in this way from any 

 function, F, satisfies the equations of equilibrium of internal stress. It is not, 

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

 the case in which p xz 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 xx , p xy and p yy 

 in each stratum must separately fulfil the conditions of equilibrium, 



d ^ _ n ^ ^ _ n 



dx Pxx + ty Pxv ~ ' dl,? xy + ~df }y] > ~ ' 



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



cPf d*f d 2 f 



Vxx ~ dy 2 ' *" - dxdy ' lhjy ~ da? ' 



where / 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 /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„ = <l>(psy), 



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 = , p yz = , we 

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



d 2 ¥ n , d 2 Y 



= and , , = , 



dx dz dy dz 



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

 ponents of stress are 



d*G d 2 Z cPG d 2 Z (<m cPG ~WG\ 2 



Vxx-f-g^ dz2 > Pyy-P -^2 dz 2 > Pm ~ P ^2 dy * dxdy ) 



d 2 G d 2 Z 



p yz = o, P , x = o, Pxy = -p d - lyl? . 



