280-282] OBLIQUE STRESSES. 509 



perpendicular to the axes of x, y, z, respectively, the three com 

 ponents of the stress, per unit area, exerted across the first of 

 these planes may be denoted by p xx , p xy ,pxz&amp;gt; respectively; those 

 of the stress across the second plane by p yx , p yy ,p yz \ and those of 

 the stress across the third plane by p Z x&amp;gt; Pzy&amp;gt; Pzz*- If we ^ x our 

 attention on an element SxSySz having its centre at P, we find, 

 on taking moments, and dividing by $xyz, 



Pyz = Pzy &amp;gt; Pzx == PXZ&amp;gt; Pxy == Pyx ( 1 )&amp;gt; 



the extraneous forces and the kinetic reactions being omitted, 

 since they are of a higher order of small quantities than the 

 surface tractions. These equalities reduce the nine components 

 of stress to six ; in the case of a viscous fluid they will also follow 

 independently from the expressions for p yz ,p z x&amp;gt;pxy in terms of the 

 rates of distortion, to be given presently (Art. 283). 



v/282. It appears from Arts. 1, 2 that in a fluid the deviation 

 of the state of stress denoted by p XX) p xy ,... from one of pressure 

 uniform in all directions depends entirely on the motion of 

 distortion in the neighbourhood of P, i.e. on the six quantities 

 a, 6, c,f, g, h by which this distortion was in Art. 31 shewn to be 

 specified. Before endeavouring to express p xx , p xy) ... as functions 

 of these quantities, it will be convenient to establish certain for 

 mulae of transformation. 



Let us draw Px t Py f , Pz in the directions of the principal 

 axes of distortion at P, arid let a , & , c be the 

 rates of extension along these lines. Further 

 let the mutual configuration of the two sets of ^ 

 axes, x, y, z and x t y , z, be specified in the y 2 , ra&amp;lt;, 

 usual manner by the annexed scheme of direc- z ! 1 3 , m 3 , n 3 . 

 tion-cosines. We have, then, 



+ * J?) 



. 



dx dy dz 



* In conformity with the usual practice in the theory of Elasticity, we reckon 

 a tension as positive, a pressure as negative. Thus in the case of a frictionless fluid 

 we have 



