AND DIAGRAMS OF FORCES, 



191 



By the rules for the composition of stress, we have for the componentH 

 of the force on this element, in terms of the six components of stress, 



X = lp x 

 Y=lp x 



+ np xl =p 

 + np,,, =p 



+ np a =p ( 

 \ 





Hence, 



(15). 



d'F d' 



ld*F d* 



d*F d* 



d"F 

 Pvz ~ P dxdy' 



~ P 



'dzdx' Px ~ p dxdy 

 By substituting these values in the equations of equilibrium 



n.v\ n.rn ri.v\ 



- = 0, &c 



....(16). 



(17), 



dx dy dz 

 it is manifest that they are fulfilled for any value of F. 



The most general solution of these equations of equilibrium is contained 

 in the values 



(18). 



dx* dz 2 



dy* 



dydz' dzdx' dxdy 



By making A = B=C=pF we get a case which, though restricted in its 

 generality, has remarkable properties with respect to diagrams of stress. We 

 have seen that a distribution of stress according to the definition above (16), 

 is consistent with itself, and will keep a body in equilibrium. Since the 

 stresses are linear functions of F, any two systems of stress can be compounded 

 by adding their respective functions, a process not applicable to the first method 

 of representation by areas. 



Let us ascertain what kind of stress is represented in this way in the 

 case of the system of cells already considered. 



