16 



The subject is further elucidated by the application of the remark- 

 able symbol e 6 ^ 1 , a symbol which in geometry serves to indicate 

 the direction in which a line is drawn with respect to a given fixed 

 line ; the same symbol is perfectly applicable as a sign of affection 

 for forces, and hence the conclusion is strengthened that the ground 

 of truth in the two sciences is the same. 



The reasoning of this paper extends not only to forces, but also 

 to velocities and moments, and to all expressions of whatever kind 

 of the pure ideas of magnitude and direction. 



If the author's reasoning be sound, the elementary propositions of 

 mechanics are necessary truths in as strict, in fact, in exactly the 

 same, sense as the elementary propositions of geometry ; and to a 

 mind which dwells upon them, the truths of the one science ought 

 to appear in as axiomatic a light as those of the other. 



April 14, 1845. 



On the Theories of the Internal Friction of Fluids in Motion, 

 and of the Equilibrium and Motion of Elastic Solids. By G. G. 

 Stokes, M.A., Fellow of Pembroke College. 



The theory of the equilibrium of fluids depends on the funda- 

 mental principle, that the mutual action of two contiguous portions 

 of a fluid is normal to the surface which separates them. This prin- 

 ciple is assumed to be true in the common theory of fluid motion. 

 But although the theory of hydrostatics is fully borne out by expe- 

 riment, there are many instances of fluid motion, the laws of which 

 entirely depend on a certain tangential force called into play by the 

 sliding of one portion of fluid over another, or over the surface of a 

 solid. The object of the first part of this paper is to form the equa- 

 tions of motion of a fluid when account is taken of this tangential 

 force, and consequently the pressure not supposed normal to the 

 surface on which it acts, nor alike in all directions. 



Since the pressure in a fluid, or medium of any sort, arises di- 

 rectly from molecular action, being in fact merely a quantity by the 

 introduction of which we may dispense with the more immediate 

 consideration of the molecular forces, and since the molecular forces 

 are sensible at only insensible distances, it follows that the pressure 

 at any point depends only on the state of the fluid in the immediate 

 neighbourhood of that point. Let the system of pressures which 

 exists about any point P of a fluid in motion be decomposed into a 

 normal pressure p, alike in all directions, due to the degree of com- 

 pression of the fluid about P, and a system S of pressures due to the 

 motion. The author assumes that the pressures belonging to the 

 system S depend only on the relative velocities of the parts of the 

 fluid immediately about P, as expressed by the nine differential co- 

 efficients of u, v and w with respect to x, y and z. [The common no- 

 tation is here employed.] He assumes, further, that the relative 

 velocities due to any arbitrary motion of rotation may be eliminated 



