306 



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



to powers of t^ integral or fractional, and we do not know but that u, v and w may be functions of 

 this nature. I do not here mention the proof which Poisson has given of the theorem in his 

 Traite de Mechanique, because it appears to me liable to an objection to which I shall presently 

 have occasion to refer : in fact, Poisson himself did not think the theorem generally true. 



It is remarkable that Mr. Power's proof, if it were legitimate, would establish the theorem 

 even when account is taken of the variation of pressure in different directions, according to the 



theory explained in Section I, if we suppose that — = 0. 



To show this we have only got to treat 



equations (12) as Mr. Power has treated the three equations of fluid motion formed on the ordinary 



hypothesis. Yet in this case the theorem is evidently untrue. Thus, conceive a mass of fluid which 



is bounded by a solid plane coinciding with the plane yz, and which extends infinitely in every 



direction on the positive side of the axis of x, and suppose the fluid at first to be at rest. Suppose 



now the solid plane to be moved in any manner parallel to the axis of y ; then, unless the solid 



plane exerts no tangential force on the fluid, (and we may suppose that it does exert some,) it 



is clear that at a given time we shall have w. = 0, « =f(jB), w = 0, and therefore udx + vdy + lodss 



will not be an exact differential. It will be interesting then to examine in this case the nature 



of the function of t which expresses the value of v. 



Supposing X, Y, Z to be zero in equations (12), and observing that in the case considered 



dp 

 we have -— = o, we set 

 dy ^ 



dv fjL (Pv 



dt p dx^ 



Differentiating this equation w - 1 times with respect to t, we easily get 



dt" ~ \p) dx^" ' 

 but when t = 0, v = when ,v > 0, and therefore for a given value of 



(24) 



all the differential 



coefficients of u with respect to t are zero. Hence for indefinitely small values of t the value of 

 u at a given point increases more slowly than if it varied ultimately as any power of t, however 

 great ; hence ti cannot be expanded in a series according to powers of t. This result' is independent 

 of the condition to be satisfied at the surface of the solid plane. 



I think what has just been proved shows clearly that Lagrange's proof of the theorem 

 considered, even with Mr. Power's improvement of it, is inadmissible. 



11. The theorem is however true, and a proof of it has been given by M. Cauchy*, which 

 appears to me perfectly free from objection, and which is very simple in principle, although it 

 depends on rather long equations, M. Cauchy first eliminates p from the three equations of 



motion by means of the conditions that 



d'p d'p 



See, he then changes the independent 



dxdy dydx 



variables from ,r, y, z, f. to a, b, c, t, where a, b, c are the initial co-ordinates of the particles. 

 The three transformed equations admit each of being once integrated with respect to t ; and 

 determining the arbitrary functions of o, 6, c by the initial values of u, v and w, the three 

 integrals have the form 



(..„' = Fill + Gw" + Hw" , &c.. 



• Memoire sur la Thiorie des Ondes, in the first volume of 

 the Memoires presentes a VInstitut. W. Cauchy has not had 

 occasion to enunciate the theorem, but it is contained in his 



equations (16). This equation may be obtained in the san.-e 

 manner in the more general case in which p is supposed to be a 

 function of p. 



