246 Prof. C. Niven on the Theory of an 



the tangential stresses be equal in pairs, so that 



*^yz =z ^zyi &zx =: &xz) ^?y — &yx» • • • v*/ 



But in the case of an imperfectly homogeneous solid this 

 specification is no longer sufficient; the stress will involve 

 other elements, depending on the form of the expression for the 

 energy of the solid due to distortion. If, for example, the 

 energy contains terms of classes (I., III.), (I., IV.), it is 

 enough to introduce nine couples of the form L ax ...L«, 

 where L uv is the couple per unit area on the face of the ele- 

 ment perpendicular to the axis of u, the axis of the couple 

 having the direction of the axis of v ; and we must also sup- 

 pose that the tangential stresses are no longer connected by the 

 relations (4). If we suppose the energy to contain terms of 

 the class (I., V.), we have also to introduce force of a different 

 type. We shall return to this point ; but in the mean time it 

 may be proved that, in order that the angular accelerations of 

 the element may not be infinite, we must have three new rela- 

 tions connecting the stress-forces and couples, as follows. 

 The symbols L w . . . L yz referring to the couples on indefinitely 

 small areas placed at the centre of the element, we may sup- 

 pose that two faces perpendicular to the axis of x are acted on 

 by couples whose axes are in this direction, of amounts equal to 



the resultant of which is the couple 



— r^ dcv dy dz. 

 dx d 



In the same way the couples on the other faces will result in 

 two couples, whose axes are in the direction of the axis of x, 

 of amounts 



— 7^- dx dy dz, —^- dx dy dz. 



We may similarly find the resultant couples in the directions 

 of the axes of y and z. But the stress-forces acting on the 

 faces give rise to the three couples 



(S yjs — $ Z y)dx dy dz, (S**— S ltz )dx dy dz, (S ^ — $ yx )dx dy dz. 



Hence the condition for finite angular acceleration requires 

 that 



