180 RECIPROCAL FIGURES, FRAMES, 



Then the equation ;>-/>* becomes 



d(dr,\ n (dldt dl 

 -3xd;)- p \dxdy Ty 



or 



Similarly, from the two other equations of equilibrium we should find 



From these three equations it follows that 



C t = 0, C, = 0, C, = 0. 



dr l _dC <K_dj[ dj[_d7 L 

 dz~dy' dx~dz' dy dx' 



and (dx + rjdy + [dz is a complete differential of some function, F, of x, y and z, 

 whence it follows that 



d_F _dF dF 



*~dx' r} ~ dy' 4 ~<fe ' 



F may be called the function of stress, because when it is known, the diagram 

 of stress may be formed, and the components of stress calculated. The form 

 of the function F is limited only by the conditions to be fulfilled at the 

 bounding surface of the body. 



The six components of stress expressed in terms of F are 



\ (#Fd*F 



} ' Pm =P \ dz> dx* ~ 



_ _ 



~ ~ P {dtf ~3 ~ (dydz) } ' Pm =P \ dz> dx* ~ (dzdx) }' Pa ~ P \dy? d'/ (dxdy 



(d'F d'F d'F d'F \ (d'F d'F d'F d'F \ 



*j ** i __ _____ ___ 1 if\ tin i rirT I 



\dzdxdxdy dx* dydz/' "" "\dxdydydz dy* dzdx/' 



(d'F d'F d'F d'F\ 

 P *~ P (dydz dzdx dz' dxdy) ' 



If -f- = z, F becomes Airy's function of stress in two dimensions, and we have 



d'F cTF d'F 



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, 



