OF FLUIDS ON THE MOTION OF PENDULUMS. [17] 



but it will be possible, in a few words, to enable the reader to form a clear idea of the meaning 



of the constant fi. 



Conceive the fluid to move in planes parallel to the plane of toy, the motion taking place 



in a direction parallel to the axis of y. The motion will evidently consist of a sort of con- 



dv 

 tinuous sliding, and the differential coefficient — may be taken as a measure of the rate of 



sliding. In the theory it is supposed that in general the pressure about a given point is com- 

 pounded of a normal pressure, corresponding to the density, which being normal is necessarily 

 equal in all directions, and of an oblique pressure or tension, altering from one direction to 

 another, which is expressed by means of linear functions of the nine differential coefficients of 

 the first order of u, v, w with respect to x, y, z, which define the state of relative motion at 

 any point of the fluid. Now in the special case considered above, if we confine our attention 

 to one direction, that of the plane of tvy, the total pressure referred to a unit of surface is 

 compounded of a normal pressure corresponding to the density, and a tangential pressure 



dv 

 expressed by u — , which tends to reduce the relative motion. 

 d% 



In the solution of equations (2), ft always appears divided by p. Let ju. = f/p. The 



constant ft! may conveniently be called the index of friction of the fluid, whether liquid or gas, 



to which it relates. As regards its dimensions, it expresses a moving force divided by the 



product of a surface, a density, and the differential coefficient of a velocity with respect to a 



line. Hence ^' is the square of a line divided by a time, whence it will be easy to adapt the 



numerical value of p' to a new unit of length or of time. 



3. Besides the general equations (2) and (3), it will be requisite to consider the equations 

 of condition at the boundaries of the fluid. For the purposes of the present paper there will 

 be no occasion to consider the case of a free surface, but only that of the common surface of 

 the fluid and a solid. Now, if the fluid immediately in contact with a solid could flow past it 

 with a finite velocity, it would follow that the solid was infinitely smoother with respect to 

 its action on the fluid than the fluid with respect to its action on itself. For, con- 

 ceive the elementary layer of fluid comprised between the surface of the solid and a 

 parallel surface at a distance h, and then regard only so much of this layer as corresponds 

 to an elementary portion dS of the surface of the solid. The impressed forces acting on 

 the fluid element must be in equilibrium with the effective forces reversed. Now conceive 

 h to vanish compared with the linear dimensions of dS, and lastly let dS vanish*. It 

 is evident that the conditions of equilibrium will ultimately reduce themselves to this, that 

 the oblique pressure which the fluid element experiences on the side of the solid must be equal 

 and opposite to the pressure which it experiences on the side of the fluid. Now if the fluid 

 could flow past the solid with a finite velocity, it would follow that the tangential pressure 



■ To be quite precise it would be necessary to say, Conceive 

 h and dS to vanish together, h vanishing compared with the 

 linear dimensions of dS; for so long as rf,S remains finite we 

 cannot suppose h to vanish altogether, on account of the curva- 



ture of the elementary surface. Such extreme precision in 

 unimportant matters tends, I think, only to perplex the reader, 

 and prevent him from entering so readily into the spirit of an 

 investigation. 



Vol. IX. Paet II. 27 



