112 Mr. stokes, on some CASES OF FLUID iMOTION. 



in which case the motion will be determined exactly. The principle explained in this article has 

 been employed in other subjects, and appears likely to be of great use in this. It is the same for 

 instance as that of successive inflitences in Electricity. 



7. If a mass of fluid be at rest or in motion in a close vessel which it entirely fills, the 

 vessel beinc either at rest or moving in any manner, any additional motion of translation com- 

 municated to the vessel will not affect the relative motion of the fluid. For it is evident that 

 on the supposition that the relative motion is not afi^ected the equation (2?) and the condition 

 (o) will still be satisfied. Also, if fjrj, sr^, ■ztsj be the components of the effective force of any 

 particle in the first case, and U, V, W, be the components of the velocity of translation, then 



(lU dV dW 



will be the components of the effective force of the same particle in the second case. Now since 

 by hypothesis nr^dx + -urjl;/ + -nrsdz is an exact differential, as follows from equations (C), and 

 U, V, W, are functions of t only, it follows at once that 



is an exact differential, where ,v, y, ;r, are the co-ordinates of any particle referred to the old axes, 

 which are themselves moving in space with velocities U, F, W. But if .t, , j/, , i-,, be the co- 

 ordinates of the same particle referred to parallel axes fixed in space, we have 



X, = 3) + jUdt, y, = y + fVdt, sr, = ^ + fWdt, 

 wiicnce, supposing the time constant, dx — dx-^, dy = dy,, dg = dz^, and therefore 



^, + — j d.., + (^, + -) dy, + (^^3 + ^^- j d., 



is an exact differential. Hence, equations (A) can be satisfied by a suitable value of p. Denoting 

 by J) the pressure about the particle whose co-ordinates are x, y, x, in the first case, the pressure 

 about the same particle in the second case will be 



IdU dV dW \ 



none of the terms of which will alter abruptly, since by hypothesis p docs not. 



Since then the present hypothesis satisfies all the requisite conditions, it follows from Art. 2 

 that that hypothesis is correct. If F be the additional effective force of any particle of the vessel 

 in consequence of the motion of translation, and we take new axes of x', y\ z' , of which the first 

 is in the direction of F, the additional term introduced into the value of the pressure will be 

 - pFx, omitting the arbitrary function of thr time. The resultant of the additional pressures on 

 the sides of the vessel will be equal to F multiplied by the mass of the fluid, and will pass 

 through the centre of gravity of the fluid, and act in the direction of — x'. 



8. Motion between two cylindrical surfaces having a common axis. 



Let us conceive a mass of fluid at rest, bounded by two cylindrical surfaces having a com- 

 mon axis, these surfaces being either infinite or bounded by two planes perpendicular to their 

 axis. Let us suppose the several generating lines of these cylindrical surfaces to be moved 

 parallel to themselves in any given manner consistent with the condition that the volume of the 

 fluid be not altered : it is required to determine the initial motion at any point of the mass. 



Since the motion will take place in two dimensions, let the fluid be referred to polar 

 co-ordinates r, 6, in a plane perpendicular to the axis, r being measured from the axis. Let 

 a be the radius of the inner surface, i that of the outer, f{9) the normal velocity of any 

 point of the inner surface, F (9) ll>e corresponding quantity for the outer. 



