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 x y,p xz , 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 zx ,p zy , Pzz*- If we nx our 

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

 on taking moments, and dividing by xyz, 



Pyz=Pzy, Pzx=Pxz, Pxy=Py* ............... (1), 



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 zx , p xy in terms of the 

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



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

 of the state of stress denoted by p xx >Pxy,-- 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, /, 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 r , Py', Pz in the directions of the principal 

 axes of distortion at P, and let a', V, 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 f , y', z', be specified in the y' 



usual manner by the annexed scheme of direc- z' 

 tion-cosines. We have, then, 



l lt m 1} n lt 

 h, m 2 , 7i 2 > 

 / 3 , m 3 , n 3 . 



-- 



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 



