An example will make this clearer. Suppose a mass of fluid to be at rest in a finite cylinder, whose axis coincides with that of z, the cylinder being entirely filled, and closed at both ends. Suppose the cylinder to be moved by impact with an initial velocity C in the direction of x; then shall

u = C, v = 0, w = 0.

For these values render uda + vdy+wdz an exact differential do, where satisfies (E); they also satisfy (a); and, lastly, the value of q obtained by integrating equations (F), namely, C' - Cpx, does not alter abruptly. But if we had supposed that was equal to Cx+C'e, where = tan1y/x, the equation (E) and the condition (a) would still be satisfied, but the value of q would be C" - p (Cx+C'0), in which the term pC'e alters abruptly from 2πρС' to 0, as 0 passes through the value 27. The condition (d) 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 correspond to a pressure which 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 condition (d), be supposed to take place in A.

It is true that equations (4) or (F), could not strictly speaking be said to hold good at those surfaces where such a discontinuity should exist. Still, to avoid the liability to error, it is well to state the condition (d) distinctly.

When the motion begins from rest, not only must udx+vdy+wdz be an exact differential do, and u, v, w, not alter abruptly, but also 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 do/dt must not alter abruptly, since otherwise p would alter abruptly from one point of the fluid to another; and dø/dt neither altering abruptly nor becoming infinite, it follows that 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 of q.

3. A result which follows at once from the principle laid down in the beginning of the 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 udx+vdy+wdz is an exact differential, it will be steady. For let u, v, w, be the initial velocities, and let us see if the velocities at the same point can remain u, v, w. First, udx+vdy+wdz being an exact differential, equations (4) will be satisfied by a suitable value of p, which value is given by equation (D). Also equation (B) is satisfied since it is so at first. The condition (a) becomes = 0, which is also satisfied since it is satisfied at first. Also the value of p given by equation (D) will not alter abruptly, for do/dt=0, or a function of t, and the velocities. dp/dx &c., are supposed not to alter abruptly. Hence, all the requisite conditions are satisfied; and hence, (Art. 2) the hypothesis of steady motion is correct*.

4. In the case of an incompressible fluid, either of infinite extent, or confined, or interrupted in any manner by any solid bodies, if the motion begin from rest, and if there be none of the cutting motion mentioned in Art. 2, the motion at the time t will be the

[N.B. It is only within a space which is at least doubly connected that such a motion is possible. Thus in the example given in the preceding article, the axis of the cylinder, where the velocity becomes infinite, may be regarded as an infinitely slender core which we are forbidden to cross, and which renders the space within the cylinder virtually ring-shaped.]

で

same as if it were produced instantaneously by the impulsive motion of the several surfaces which bound the fluid, including among these surfaces those of any solids which may be immersed in it. For let u, v, w, be the velocities at the time t. Then by a known theorem udx+vdy+wdz will be an exact differential dø, and will not alter abruptly (Art. 2). 4 must also satisfy the equation (E), and the conditions (a) and (b). Now if u', v', w', be the velocities on the supposition of an impact, these quantities must be determined by precisely the same conditions as u, v and w. But the problem of finding u', v' and w', being evidently determinate, it follows that the identical problem of finding u, v and w is also determinate, and therefore the two problems have the same solution; so that

u = u', v=v', w=w'.

This principle has been mentioned by M. Cauchy, in a memoir entitled Mémoire sur la Théorie des Ondes, in the first volume of the Mémoires des Savans Étrangers (1827), page 14. It will be employed in this paper to simplify the requisite calculations by enabling us to dispense with all consideration of the previous motion, in finding the motion of the fluid at any time in terms of that of the bounding surfaces. One simple deduction from it is that, when all the bounding surfaces come to rest, each element of the fluid will come to rest. Another is, that if the velocities of the bounding surfaces are altered in any ratio the value of will be altered in the same ratio.

5. Superposition of different motions.

In calculating the initial motion of a fluid, corresponding to given initial motions of the bounding surfaces, we may resolve the latter into any number of systems of motions, which when compounded give to each point of each bounding surface a velocity, which when resolved along the normal is equal to the given velocity resolved along the same normal, provided that, if the fluid be enclosed on all sides, each system be such as not to alter its volume. For let u', v', w', v', o', be the values of u, v, &c., corresponding to the first system of motions; u", v", &c., the values of those quantities corresponding to the second system, and so on; so that

u = u' + u" + ..., v = v' + v"'+..., w = w+w" + ...,

v = v' + v" + ..., σ = o' + σ" + ....

Then since we have by hypothesis u'da + v'dy + w'dz an exact differential do', u"dx + v"dy+w"dz an exact differential do", and so on, it follows that udx + vdy+wdz is an exact differential. Again by hypothesis 'o', v' =σ", &c., whence vσ. Also, if the fluid extend to an infinite distance, u, v, and w must there vanish, since that is the case with each of the systems u', v', w', &c. Lastly, the quantities p', d", &c., not altering abruptly, it follows that , which is equal to '+"+ ..., will not alter abruptly. Hence the compounded motion will satisfy all the requisite conditions, and therefore (Art. 2) it is the actual motion.

It will be observed that the pressure p will not be obtained by adding together the pressures due to each of the above systems of velocities. To find p we must substitute the complete value of in equation (D). If, however, the motion be very small, so that the square of the velocity is neglected, it will be sufficient to add together the several pressures just mentioned.

In general the most convenient systems into which to decompose the motion of the bounding surfaces are those formed by considering the motion of each surface, or of a certain portion of each surface, separately. Such a portion may be either finite or infinitesimal. In fact, in some of the cases of motion that will be presently given, where is expressed by a double integral with a function under the integral sign expressing the motion of the bounding surfaces, it will be found that each element of the integral gives a value of 4 such that, except about the corresponding element of the bounding surface, the motion of all particles in contact with those surfaces is tangential.

A result which follows at once from this principle, and which appears to admit of comparison with experiment, is the following. Conceive an ellipsoid, or any body which is symmetrical with respect to three planes at right angles to each other, to be made. to oscillate in a fluid in the direction of each of its three axes in succession, the oscillations being very small. Then, in each case, as may be shewn by the same sort of reasoning as that employed in Art. 8, in the case of a cylinder, the effect of the inertia of the fluid will be to increase the mass of the solid by a mass having a certain unknown ratio to that of the fluid displaced. Let the axes of co-ordinates be parallel to the axes of the solid; let x, y, z, be

the co-ordinates of the centre of the solid, and let M, M', M", be the imaginary masses which we must suppose added to that of the solid when it oscillates in the direction of the axes of x, y, z, respectively. Let it now be made to oscillate in the direction of a line making angles a, B, y, with the axes, and let s be measured along this line. Then the motions of the fluid due to the motions of the solid in the direction of the three axes will be superimposed. The motion being supposed to be small, the resultant of the pressures of the fluid on the solid will be three forces, equal to

respectively, in the directions of the three axes.

these in the direction of the motion will be M, d's/dt where

M = M cos2 a + M' cos3ß + M' cosy.

Each of the quantities M, M', M" and M,, may be determined by observation, and we may find whether the above relation holds between them. Other relations of the same nature may be deduced from the principle explained in this article.

6. Reflection.

Conceive two solids, A and B, immersed in a fluid of infinite extent, the whole being at rest. Suppose A to be moved in any manner by impulsive forces, while B is held at rest. Suppose the solids A and B of such forms that, if either were removed, and the several points of the surface of the other moved instantaneously in any given manner, the motion of the fluid could be determined: then the actual motion can be approximated to in the following manner. Conceive the place of B to be occupied by fluid, and A to receive its given motion; then by hypothesis the initial motion of the fluid can be determined. Let the velocity with which the fluid in contact with that which is supposed to occupy B's place penetrates into the latter be found, and then suppose that the several points of the surface of B are moved with normal velocities equal and opposite to those just found, A's place being supposed to be occupied by fluid. The motion of the fluid corresponding to the velocities of the several points of the surface of B can then be found, and A must now be treated as B has been, and so on. The system of velocities of the particles of the fluid corresponding to