Mr. stokes, on SOME CASES OF FLUID MOTIOX. 109 



An example will make this clearer. Suppose a mass of fluid to be at rest in a finite cylinder, 

 whose axis coincides witli that of r, the cylinder being entirely filled, and closed at both ends. Sup- 

 pose the cylinder to be moved by impact with an initial velocity C in the direction of .r ; then shall 



u = C, t) = 0, w = 0. 



For these values render tidx + vdy + wd« an exact differential d(p, where <p satisfies {E)\ they also 

 satisfy (o) ; and, lastly, the value of q obtained by integrating equations (F), namely, C' - Cp-v, 



does not alter abruptly. But if we had supposed that (p were equal to Cx + C^6, where — tan"' — , 



the equation (E) and the condition (a) would still be satisfied, but the value of q would be 

 C" - p{C.v + C 9), in which the term pC'9 alters abruptly from ZirpC to 0, as 6 passes through 

 the value 2 7r. The condition (rf) then alone shews that the former and not the latter is the true 

 solution of the problem. 



The fact that the analytical conditions of a problem in fluid motion, as far as those conditions 

 depend on the velocities, may be satisfied by values of those velocities, which notwithstanding corre- 

 spond to a pressure wliich alters abruptly, may be thus explained. Conceive two masses of the same 

 fluid contained in two similar and equal close vessels A and B. For more simplicity, suppose these 

 vessels and the fluid in them to be at first at rest. Conceive the fluid in B to be divided by an 

 infinitely thin lamina which is capable of assuming any form, and, at the same time, of sustaining 

 pressure. Suppose the vessels A and B to be moved in exactly the same manner, the lamina in 

 B being also moved in any arbitrary manner. It is clear that, except for one particular motion of 

 the lamina, the motion of the fluid in B will be different from that of the fluid in A. The velocities 

 u, V, w, will in general be different on opposite sides of the lamina in B. For particular motions of 

 the lamina however the velocities u, v, w, may be the same on opposite sides of it, while the 

 pressures are different. The motion which takes place in B in this case might, only for the con- 

 dition ((/), be supposed to take place in A. 



It is true that equations (A) or (F), could not strictly speaking be said to hold good at those 

 surfaces where such a discontinuity should exsist. Still, to avoid the liability to error, it is well to 

 state the condition (rf) distinctly. 



When the motion begins from rest, not only must ud<v + vdy + wdz be an exact differential dcp, 

 and n, v, « , not alter abruptly, but also (p must not alter abruptly, provided the particles in 

 contact with the several surfaces remain in contact with those surfaces; for if this condition be not 

 fulfilled, the surface for which it is not fulfilled will as it were cut the fluid into two. For it follows 



from the equation (D) that —- must not alter abruptly, since otherwise p would alter abruptly 



from one point of the fluid to another; and — ^ neither altering abruptly nor becoming infinite, it 



follows tiiat cj) will not alter abruptly. Should an impact occur at any period of the motion, it 

 follows from equations (F) that that cannot cause the value of (p to alter abruptly, since such an 

 abrupt alteration would give a corresponding abrupt alteration in the value off/. 



3. A result which follows at once from the principle laid down in the beginning of tlie last article 

 is this, that when the motion of a fluid in a close vessel which is at rest, and is completely filled, is 

 of such a kind that wrf.B + vdy + wdz is an exact differential, it will be steady. For let u, v, iv, be 

 tiie initial velocities, and let us see if the velocity at tlie same point can remain u, v, w. First, 

 ndx + rdy+ wdz being an exact differential, equations {A) will be satisfied by a suitable value of />, 

 which value is given by e(iuation (Z>). Also ecpiatiDU {B) is .satisfied since it is so at first. Tiic 

 condition («) becomes i- = 0, which is also satisfied since it is satisfied at first. Also the value of p 



Riven by e(iuation {]>) will not alter abruptly, for ~ = 0, or a function of ^ and the velocities -- &c., 



dt die 



