AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 289 



SECTION I. 



Explanation of the Theory of Fluid Motion proposed. Formation of the Differential 

 Equations. Application of these Equations to a few simple cases. 



1. Before entering on the explanation of this theory, it will be necessary to define, or fix 

 the precise meaning of a few terras which I shall have occasion to employ. 



In the first place, the expression "the velocity of a fluid at any particular point" will require 

 some notice. If we suppose a fluid to be made up of ultimate molecules, it is easy to see that these 

 molecules must, in general, move among one another in an irregular manner, through spaces 

 comparable with the distances between them, when the fluid is in motion. But since there 

 is no doubt that the distance between two adjacent molecules is quite insensible, we may neglect the 

 irregular part of the velocity, compared with the common velocity with which all the molecules 

 in the neighbourhood of the one considered are moving. Or, we may consider the mean velocity 

 of the molecules in the neighbourhood of the one considered, apart from the velocity due to 

 the irregular motion. It is this regular velocity which I shall understand by the velocity of 

 a Jluid at any point, and I shall accordingly regard it as varying continuously with the 

 co-ordinates of the point. 



Let P be any material point in the fluid, and consider the instantaneous motion of a very 

 small element E of the fluid about P. This motion is compounded of a motion of translation, 

 the same as that of P, and of the motion of the several points of E relatively to P. If we 

 conceive a velocity equal and opposite to that of P impressed on the whole element, the remaining 

 velocities form what I shall call the relative velocities of the points of the fluid about P; and 

 the motion expressed by these velocities is what I shall call the relative motion in the neigh- 

 bourhood of P. 



It is an undoubted result of observation that the molecular forces, whether in solids, liquids, 

 or gases, are forces of enormous intensity, but which are sensible at only insensible distances. 

 Let E' be a very small element of the fluid circumscribing E, and of a thickness greater than 

 the distance to which the molecular forces are sensible. The forces acting on the element E 

 are the external forces, and the pressures arising from the molecular action of E'. If the 

 molecules of E were in positions in which they could remain at rest if E were acted on by no 

 external force and the molecules of E' were held in their actual positions, they would be in 

 what I shall call a state of relative equilibrium. Of course they may be far from being in a 

 state of actual equilibrium. Thus, an element of fluid at the top of a wave may be sensibly in 

 a state of relative equilibrium, although far removed from its position of equilibrium. Now, in 

 consequence of the intensity of the molecular forces, the pressures arising from the molecular action 

 on E will be very great compared with the external moving forces acting on E. Consequently 

 the state of relative equilibrium, or of relative motion, of the molecules of E will not be sensibly 

 affected by the external forces acting on E. But the pressures in different directions about 

 the point P depend on that state of relative equilibrium or motion, and consequently will not 

 be sensibly affected by the external moving forces acting on E. For the same reason they will not 

 be sensibly affected by any motion of rotation common to all the points of E ; and it is a direct 

 consequence of the .second law of motion, that they will not be affected by any motion of translation 

 common to the whole element. If the molecules of E were in a state of relative equilibrium, 

 the pressure would be equal in all directions about P, as in the case of fluids at rest. Hence 

 I shall assume the following principle : — 



That the difference between the pressure on a plane in a given direction passing through 



any point P of a Jluid in motion and the pressure which would etrist in all directions 



about P if the fluid in its neighbourhood were in a state of relative equilibrium depends 



only on the relative motion of the Jluid immediately about P ; and that the relative motion 



Vol. VIII. Paet III. P e 



