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



U =V COS A'x + V COS xy + W cos wz, 

 dx = cos.rxrfx' + cos xydy' + cos xzdz, &c. ; 



whence, taking account of the well known relations between the cosines involved in these equations, 

 we easily find 



Udx + Vdy + Wdss' = Udx' + Vdy' + Wdz'. 



It appears therefore that the relative velocities U, V, W, which remain after eliminating a certain 

 angular velocity, are such that Udx' + Vdy' + Wdz' is ultimately an exact differential. Hence 

 the values of U, V, W are the same as would have been obtained from equations (2) applied 

 directly to the new axes, whence the truth of the proposition enunciated at the head of this 

 paragraph is manifest. 



The motion corresponding to the velocities f7, V^, IF may be further decomposed into a 

 motion of dilatation, positive or negative, which is alike in all directions, and two motions which I 

 shall call motions of shifting, each of the latter being in two dimensions, and not affecting the 

 density. For let 3 be the rate of linear extension corresponding to a uniform dilatation ; let crx', 

 - ay'^ be the velocities parallel to ,r , y^, corresponding to a motion of shifting parallel to the 

 the plane x^y^, and let cr' x', — <t'»/ be the velocities parallel to x^, z^, corresponding to a similar 

 motion of shifting parallel to the plane xz^. The velocities parallel to x^, y, %^ respectively 

 corresponding to the quantities ^, a and a' will be (^ + <t + a) x'^, (^ — a) y', (^ — a') x ', and 

 equating these to f/, F, W^ we shall get 



^ = 1 (e' + e" + e'"), o- = ^ (e + e'"- 2 e"), o-'= ^ (e'+ e"- 2e"')- 

 Hence the most general instantaneous motion of an elementary portion of a fluid is compounded 

 of a motion of translation, a motion of rotation, a motion of uniform dilatation, and two motions of 

 shifting of the kind just mentioned. 



3. Having determined the nature of the most general instantaneous motion of an element 

 of a fluid, we are now prepared to consider the normal pressures and tangential forces called 

 into play by the relative displacements of the particles. Let p be the pressure which would exist 

 about the point P if the neighbouring molecules were in a state of relative equilibrium : let p + p^ 

 be the normal pressure, and t^ the tangential action, both referred to a unit of surface, on a plane 

 passing through P and having a given direction. By the hypotheses of Art. 1. the quantities p_, t^ 

 will be independent of the angular velocities lo', to", w", depending only on the residual relative 

 velocities U,V,W, or, which comes to the same, on e, e" and e'", or on o-, <r' and 3. Since this residual 

 motion is symmetrical with respect to the axes of extension, it follows that if the plane considered 

 is perpendicular to any one of these axes the tangential action on it is zero, since there is no more 

 reason why it should act in one direction rather than in the opposite ; for by the hypotheses 

 of Art. 1. the change of density and temperature about the point P is to be neglected, the 

 constitution of the fluid being ultimately uniform about that point. Denoting then by p+p', 

 p + p", p + p" the pressures on planes perpendicular to the axes of .r?^, y,, z^, we must have 



p'= (e, e", e'"), p"= <^ (e", e'\ e), p"'= <f> (e'", e, e"), 



<p (e',e",e"') denoting a function of e, e" and e" which is symmetrical with respect to the two 

 latter quantities. The question is now to determine, on whatever may seem the most probable 

 hypothesis, the form of the function (p. 



Let us first take the simpler case in which there is no dilatation, and only one motion of 

 shifting, or in which e' = - e, e" = 0, and let us consider what would take place if the 

 fluid consisted of smooth molecules acting on each other by actual contact. On this supposition, 

 it is clear, considering the magnitude of the pressures acting on the molecules compared with 

 their masses, that they would be sensibly in a position of relative equilibrium, except when 

 the equilibrium of any one of them became impossible from the displacement of the adjoining 



