296 Mr. stokes, on THE FRICTION OF FLUIDS IN MOTION, 



It may also be very easily proved directly that the value of sS, the rate of cubical dilatation, 

 satisfies the equation 



. du dv dw 



^^ = :r +7-+ T O) 



dx dy dz 



Let Pj, {ytz), (j/tx) be the quantities referring to the axis of y, and Pj, {xtx), (zty) those 

 referring to the axis of z, which correspond to P, &c. referring to the axis of x. Then we see 

 that iytz) = (zty), (ztx) = (xtz), {xty) = {ytx). Denoting these three quantities by T,, T^, 7',, 

 and making the requisite substitutions and interchanges, we have 



"■-"-'-£-*)■ 



dy 

 dw 





.(8) 



_, Idv dw\ idw du\ (du dv\ 



^'--''[dz^Ty)' ^'^-'^IdT+d^J' ^^^-''[d^'-dij- 

 It may also be useful to know the components, parallel to .r, y, z, of the oblique pressure on a 

 plane passing through the point P, and having a given direction. Let I, m, n be the cosines of the 

 angles which a normal to the given plane makes with the axes of x, y, z; let P, Q, R be the 

 components, referred to a unit of surface, of the oblique pressure on this plane, P, Q, R being 

 reckoned positive when the part of the fluid in which is situated the normal to which I, m and n 

 refer is urged by the other part in the positive directions of x, y, z, when /, m and n are positive. 

 Then considering as before a tetrahedron of which the base is parallel to the given plane, the 

 vertex in the point P, and the sides parallel to the co-ordinate planes, we shall have 



P = IP^ + mT^ + nTi, 



:1 

 T 



Q = lT^ + mP^ + nT„} (9) 



R = lT^ + mTt+ nP^. 



In the simple case of a sliding motion for wrhich m = 0, v =f(x), w = 0, the only forces, 

 besides the pressure p, which act on planes parallel to the co-ordinate planes are the two tangential 



forces 7^3, the value of which in this case is - ;u — . In this case it is easy to show that the axes of 



dw 



extension are, one of them parallel to Oz, and the two others in a plane parallel to xy, and inclined 



at angles of 45° to Ox. We see also that it is necessary to suppose /u. to be positive, since 



otherwise the tendency of the forces would be to increase the relative motion of the parts of tiie 



fluid, and the equilibrium of the fluid would be unstable. 



5. Having found the pressures about the point P on planes parallel to the co-ordinate planes, 

 it will be easy to form the equations of motion. Let JT, Y, Z be the resolved parts, parallel 

 to the axes, of the external force, not including the molecular force ; let p be the density, t the 

 time. Consider an elementary parallelepiped of the fluid, formed by planes parallel to the 

 co-ordinate planes, and drawn through the point (x, y, z) and the point {x + Ax, y + Ay, z + Az). 

 The mass of this element will be ultimately pAx Ay Az, and the moving force parallel to x arising 

 from the external forces will be ultimately pX Ax Ay Az ; the effective moving force parallel 



JDu 



to .f will be ultimately p -— Ax Ay Az, where D is used, as it will be in the rest of this paper. 



