294 Mr. stokes, ON THE FRICTION OF FLUIDS IN MOTION, 



Hence if ^ = 0, i. e. if e + e" + e" = 0, we shall have in general 



p' = - 2/ie', p" = - 2/ne", p" = - Smb'" (5) 



It was by this hypothesis of starts that I first arrived at these equations, and the differential 

 equations of motion which result from them. On reading Poisson's memoir however, to which 

 I shall have occasion to refer in Section iv., I was led to reflect that however intense we may 

 suppose the molecular forces to be, and however near we may suppose the molecules to be to their 

 positions of relative equilibrium, we are not therefore at liberty to suppose them in those positions, 

 and consequently not at liberty to suppose the pressure equal in all directions in the intervals of 

 time between the starts. In fact, by supposing the molecular forces indefinitely increased, 

 retaining the same ratios to each other, we may suppose the displacements of the molecules from 

 their positions of relative equilibrium indefinitely diminished, but on the other hand the force of 

 restitution called into action by a given displacement is indefinitely increased in the same proportion. 

 But be these displacements what they may, we know that the forces of restitution make equilibrium 

 with forces equal and opposite to the effective forces ; and in calculating the effective forces we 

 may neglect the above displacements, or suppose the molecules to move in the paths in which they 

 would move if the shifting motion took place with indefinite slowness. Let us first consider a 

 single motion of shifting, or one for which e" = - e', e" = 0, and let j>^ and t^ denote the same 

 quantities as before. If we now suppose e' increased in the ratio of m to 1, all the effective forces 

 will be increased in that ratio, and consequently p^ and t^ will be increased in the same ratio. We 

 may deduce the values of p, p" and p'" just as before, and then pass by the same reasoning to 

 the case of two motions of shifting which coexist, only that in this case the reasoning will be demon- 



strative, since we may suppose the time — indefinitely diminished. If we suppose the state of 



things considered in this paragraph to exist along with the motions of starting already considered, 

 it is easy to see that the expressions for p', p" and p'" will still retain the same form. 



There remains yet to be considered the effect of the dilatation. Let us first suppose the 

 dilatation to exist without any shifting : then it is easily seen that the relative motion of the 

 fluid at the point considered is the same in all directions. Consequently the only effect which 

 such a dilatation could have would be to introduce a normal pressure p., alike in all directions, in 

 addition to that due to the action of the molecules supposed to be in a state of relative equilibrium. 

 Now the pressure p^ could only arise from the aggregate of the molecular actions called into play 

 by the displacements of the molecules from their positions of relative equilibrium ; but since these 

 displacements take place, on an average, indifferently in all directions, it follows that the actions 

 of which p^ is composed neutralize each other, so that p = 0. The saaie conclusion might be 

 drawn from the hypothesis of starts, supposing, as it is natural to do, that each start calls into 

 action as much increase of pressure in some directions as diminution of pressure in others. 



If the motion of uniform dilatation coexists with two motions of shifting, I shall suppose, 

 for the same reason as before, that the effects of these different motions are superimposed. Hence 

 subtracting 3 from each of the three quantities e, e' and e'", and putting the remainders in the 

 place of e, e" and e" in equations (5), we have 



p = |/z(e" + e" - 2e'), p" = |(n(e"' + e - 2e"), p'" = |,i(e' + e" - 2e"') (fi) 



If we had started with assuming (p(e', e", e") to be a linear function of e, e" and e", 

 avoiding all speculation as to the molecular constitution of a fluid, we should have had at once 

 p' = Ce + C'{e" + e'"), since p' is symmetrical with respect to e" and e" ; or, changing the 

 constants, p' = |-/ji(e" + e" - 2e') + k (e' + e" + e"). The expressions for p" and p'" would be 

 obtained l)y interchanging the requisite quantities. Of course we may at once put (c = if we 

 assume that in the case of a uniform motion of dilatation the pressure at any instant depends 

 only on the actual density and temperature at that instant, and not on the rate at which the 



