RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 17 



Similarly, for the moments about the axes of y and z, we obtain the equa- 

 tions 



Pzx = Vx* and P*v — iV • 



Now, let us assume for the present 



§ = B,-C„|=A,, § = B 1 + C„ 



| = B ! + C ! ,| = B 1 -C J , § = A S . 



Then the equation ^ = p zy becomes 



/d^&n d^_^n\ _ (WML _ dXdJL\ 



P \dz dx dx dz) ~ P \dx dy dy dx) 

 or 



(B 2 - C 2 ) (B, - C 3 ) - A, (B, + C x ) = (B 2 + C 2 ) (B 3 + 0.) - A, (B x - C,) 



= A X G X + B 3 C 2 + B 2 C 3 . 



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



= A 2 C 2 + B 1 C 3 + B 3 C 1 

 = A 3 C 3 + B 2 C X + B^ . 



From these three equations it follows that 



Hence 



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

 whence it follows that 



dx dy dz 



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 bound- 

 ing surface of the body. 



The six components of stress expressed in terms of F are 



/cPFd?F /d 2 F\ 2 \ _ /d?F<£F (d 2 F\ 2 \ _ /d 2 F dF 2 ( d 2 F\ 2 \ 



p xx -Py dy2 dz 2 \dydz) ) ' Pyy ~ P \dz 2 dx 2 \dzdx) } Pzz ~ p \ dx 2 dy 2 \dxdy) ) ' 



/ d 2 F d 2 F ^ 2 F^ 2 F\ _ / d 2 F d 2 F d 2 F d 2 F \ _ /d 2 F d 2 F d 2 F d 2 F\ 

 yi ^\dzdxdxdy dx 2 dydz)' '^ zx ~ P \dxdydydz dy 2 dzdx)' Pxy ~^\dydzdzdx dz 2 dxdy) ' 

 VOL. XXVI. PART I. E 



c x = o 



c 2 = o 



c 3 = o. 



dr) _ dt, 



dz dy ' 



d£ _ d% 



dx ~ dz 



d% _ dr) 



dy ~ dx 



