26 



MR CLERK MAXWELL ON 



Hence this quantity depends only on the position of the point, and not on the 

 directions of s 1 s 2 s 3 or of x y z, let us call it A 2 F. 



Now, let us take an element of area perpendicular to s, and let us suppose 

 that the stress on this element is compounded of a normal pressure = pA 2 F, 



and a tension parallel to o- and equal to p - . 



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

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



X = lp xx + mp xy + tvpua — pi ZA 2 F — \- j 



Y = Ip X y + Vipyy + «£>„« = pltll/^F ~ f^ ) 



Z = lp xz + mp,, z + wp a = p>( «A 2 F — v- J 



(15). 



Hence, 



p xx = P ( A 2 F - £) = p ( A'F - ^ = P (^ + ^ 



/ f / 2 F d 2 F\ /^ 2 F d 2 F\ /d 2 F d-Y\ 



(16). 



P* = -P 



d 2 F 



p !X = -p 



d 2 F 



P*rP : 



d 2 F 



dxdy ' rzx r dzdx ' rxyr dxdy 



By substituting these values in the equations of equilibrium 



dp, 



XX + dpxy + dpxZ_ _ Q &(;i 



dx dy dz 



(17), 



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 



P* 



d 2 B d?C 



dz 2 dy 2 



Pyy = 



d 2 G d?A 

 dx 2 dz 2 



Pvz=~ 



d 2 A 



dydz 



Pzx — 



d 2 B 



dzdx 



Pzz = 



Pa 



d 2 A d 2 B 

 dy 2 dx 2 



d 2 G 

 dxdy 



(18). 



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. 



Since F in each cell is a linear function of x, y, z, there can be no stress at any 



