170] 



NATURE OF STRESS. 



449 



Introducing the values of 

 Y nj Z n from 86) this becomes 



, $Z from equations 90) and of 



92) / / {y [Z x cos (nx) -f Z y cos (ny) -f Z* cos (w0)] 



- 0[Y X cos (w#) -f Yj, cos (ny) -f F, cos (ws)]} d$ 

 *SL 



(^ 

 \0aj 



+-*+ - 



Writing the term 



and 



and applying the divergence theorem, all the surface integrals cancel 

 each other and there remains only the volume integral 



93) 



As before, the field of integration being arbitrary, the integrand 

 must vanish, and we obtain, after applying the same process to the 

 remaining two equations, 



QA\ v 7 7 v ~y v 



y) JL z = Zy y , ZJ X = jC^ Z j J^y =F L x . 



We may also obtain these equations by considering the stresses 

 on the faces of an infinitesimal 

 cube (Fig. 147). We shall denote 

 the tangential components or 

 shearing stresses 94) by T x , T y , T z , 

 the normal components or trac- 

 tions by P XJ P y , P z . The stress 

 at any point is determined in 

 terms of these six components, 

 for we may find the stress -vector 

 F n , whose direction - cosines . are 

 u'j P'> ?' f r an J stress plane 

 whose normal has the direction 

 cosines a, fl, y by equations 86), 

 which in our present notation 



Fig. 147. 



become 



95) 



X n = F n a' = 

 Y n = F n p = 



TyK 



WEBSTER, Dynamics. 



29 



