216 



VISCOSITY. 



[CHAP. ix. 



To form the equations of motion let us take, as in Art. 6, an 

 element dxdydz having its centre at P, and resolve the forces 

 acting on it parallel to x. Taking first the pair of faces perpen 

 dicular to x, the difference of the tensions on these parallel to 



j XX dx.dydz. From the other two pairs we obtain 

 else 



dz . dxdy y respectively. Hence, with our usual 



x will be - 



j~dy .dzdx and 

 dy 9 dz 



notation, 



and, similarly, 



.(2). 



176. It appears from Arts. 2, 3 that the deviation of the 

 state of stress denoted by p xx ,p xv , &c. from one of uniform pres 

 sure in all directions depends entirely on the motion of distortion 

 of the fluid in the neighbourhood of P, i.e. on the quantities 

 a, &, c,/, g, h by which this distortion was in Art. 38 shewn to be 

 specified. Before endeavouring to express^., p xy , &c. as functions 

 of these quantities, it is convenient to establish certain formulas 

 of transformation. 



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

 axes of distortion at P, and let a , 5 , 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, y , z, be specified in the usual 

 manner by the annexed scheme of direction-cosines. 



x y z 



We have, then, 



du f -, d 



dx 



+ l.v + Ijaf) 



