458 



ON PHYSICAL LINES OF FORCE. 

 1 



If we write 

 then 



\ = vm ) and y = 

 l^ 



J_ 

 1 



(2), 



PROP. III. To find the resultant force on an element of the medium, 

 arising from the variation of internal stress. 



We have in general, for the force in the direction of x per unit of volume 

 by the law of equilibrium of stresses*, 



In this case the expression may be written 

 1 (dJM da d 



da 



Remembering that a j~ 



da 



= j~ (<*' + /? + /), this becomes 



1 (dfi da\ 1 (da dy\ dp, 



The expressions for the forces parallel to the axes of y and z may be written 

 down from analogy. 



* Rankine's Applied Mechanics, Art. 116. 



