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



constructed, can hardly be considered as affording us any material insight into the laws of nature ; 

 nor will they enable us to pass from the consideration of the phenomena from which they were 

 derived to that of others of a different class, although depending on the same causes. 



In reflecting on the principles according to which the motion of a fluid ought to be calculated 

 when account is taken of the tangential force, and consequently the pressure not supposed the 

 same in all directions, I was led to construct the theory explained in the first section of this 

 paper, or at least the main part of it, which consists of equations (13), and of the principles 

 on which they are formed. I afterwards found that Poisson had written a memoir on the same 

 subject, and on referring to it I found that he had arrived at the same equations. The method 

 which he employed was however so different from mine that I feel justified in laying the latter 

 before this Society*. The leading principles of my theory will be found in the hypotheses of 

 Art. I, and in Art. 3. 



The second section forms a digression from the main object of this paper, and at first sight 

 may appear to have little connexion with it. In this section I have, I think, succeeded in shewing 

 that Lao-range's proof of an important theorem in the ordinary theory of Hydrodynamics is 

 untenable. The theorem to which I refer is the one of which the object is to show that 

 udx + vdy + wdx, (using the common notation,) is always an exact differential when it is so 

 at one instant. I have mentioned the principles of M. Cauchy's proof, a proof, I think, liable 

 to no sort of objection. I have also given a new proof of the theorem, which would have served to 

 establish it had M. Cauchy not been so fortunate as to obtain three first integrals of the general 

 equations of motion. As it is, this proof may possibly be not altogether useless. 



Poisson, in the memoir to which I have referred, begins with establishing, according to 

 his theory, the equations of equilibrium and motion of elastic solids, and makes the equations of 

 motion of fluids depend on this theory. On reading his memoir, I was led to apply to the theory 

 of elastic solids principles precisely analogous to those which I have employed in the case of 

 fluids. The formation of the equations, according to these principles, forms the subject of 

 Sect. III. 



The equations at which I have thus arrived contain two arbitrary constants, whereas Poisson's 

 equations contain but one. In Sect. iv. I have explained the principles of Poisson's theories of 

 elastic solids, and of the motion of fluids, and pointed out what appear to me serious objections 

 against the truth of one of the hypotheses which he employs in the former. This theory seems 

 to be very generally received, and in consequence it is usual to deduce the measure of the cubical 

 compressibility of elastic solids from that of their extensibility, when formed into rods or wires, 

 or from some quantity of the same nature. If the views which I have explained in this section 

 be correct, the cubical compressibility deduced in this manner is too great, much too great in 

 the case of the softer substances, and even the softer metals. The equations of Sect. iii. have, 

 I find, been already obtained by M. Cauchy in his Exercises Mathematiqiies, except that he 

 has not considered the eftect of the heat developed by sudden compression. The method which 

 I have employed is different from his, althouglj in some respects it much resembles it. 



The equations of motion of elastic solids given in Sect. lu. are the same as those to which 

 different authors have been led, as being the equations of motion of the luniiniferous ether in 

 vacuum. It may seem strange that the same equations should have been arrived at for cases 

 so different ; and I believe this has appeared to some a serious objection to the employment of 

 those equations in the case of light. I think the reflections which I have made at the end of 

 Sect. IV., where I have examined the consequences of the law of continuity, a law which seems 

 to pervade nature, may tend to remove the difficulty. 



• The same equations have also been obiained by Navier ] T. vi.) but his principles dift'er from mine still more than do 

 in the case of an incompressible fluid, {Mem. de I'lnstilut, \ Poisson's. 



