310 Prof. Challis on the received principles of Hydrodynamics. 



Let p be the pressure, p the density, and u, v, w the resolved 

 parts of the velocity in the directions of rectangular axes in any 

 perfect fluid at any point whose coordinates are x, y, z at the 

 time t. Then regarding it as an axiom that the state of the fluid 

 at all points and at all times admits of being analytically ex- 

 pressed, the complete solution of every hydrodynamical problem 

 will conduct to expressions for the five quantities p } p, u, v, w y 

 inasmuch as, when these have been obtained as functions of co- 

 ordinates and the time, the pressure, density, velocity, and direc- 

 tion of the velocity are determined for each point at each instant. 

 Consequently the solution will consist of five equations, such as 



P =/i fo V> *> t)> P =/sfe V> s>t), u =f 3 {x, y t z> t), 



v =f 4 (x, y, z>t)y w. =/ 5 0, y, z, t) . 



The elimination of x } y, z, and t from these equations would con- 

 duct to an equation of condition, F (p, p t u, v, w) — 0, between 

 the five quantities. This result proves that in the solution of 

 every hydrodynamical problem there exists between the quanti- 

 ties determined a relation the analytical expression of which 

 does not contain explicitly the coordinates and the time. 



Now the condition thus shown by a priori considerations to be 

 necessary can be satisfied by an arbitrary relation between the 

 quantities^ — that is, by one which is independent of the particular 

 problem. For by means of such a relation it will be analytically 

 possible to eliminate from the general hydrodynamical equations 

 one of the unknown quantities, as, for instance, p. Then in any 

 particular instance the integration of the equations gives the 

 values of the other four as functions of the coordinates and the 

 time. These being found, the value of the first is obtainable 

 from the assumed relation. By this process the state of the fluid, 

 subject to the arbitrary condition, will be ascertained, in the pro- 

 posed instance, for all points at all times. The arbitrary con- 

 dition has the effect of defining the fluid ; and evidently the 

 number of different kinds of fluid is unlimited. 



It having been thus proved to be allowable to assume in hy- 

 drodynamics an arbitrary relation between p, p, u, v, w, let us 

 suppose, for illustration, that 



p = Ap« + W + (V + Dw s , 



the coefficients A, B, C, D and the indices a, ft y, 8 being con- 

 stant and arbitrary. By this equation^ might be eliminated 

 from the general hydrodynamical equations, so that the integra- 

 tions of the equations for a particular problem would give ex- 

 pressions for p, u, v, w as functions of the coordinates and the 

 time, whence that for p might be obtained by means of the above 

 equality. As the coefficients and the indices are wholly arbitrary 



