the Vibrations of an Elastic Fluid. 139 



Then 



, df . df A<b 



and the required conditions are satisfied if, where x = and y = } 



7/> 7/* 



we have also — =0 and ■— = 0, and consequently the value of/ 



a maximum or minimum. If / be a maximum, the velocity 

 estimated parallel to the axis of z is also a maximum along that 

 axis ; and since . this, as we shall afterwards see, must be the 

 case, the minimum value is excluded. 



21. The next step in the investigation of the character of the 

 free motion of the fluid is to substitute the above values of u, v, 

 w in the two general equations (1) and (2) in art. 3, and to 

 eliminate p between them. The equation resulting from this 

 elimination is given under Prop. X., in a communication "On 

 the Principles of Hydrodynamics " in the Philosophical Maga- 

 zine for December 1852. As, however, the condition of the 

 integrability of udx + vdy + wdz applies only to points inimedi- 

 diately contiguous to the axis of z, we must omit in that equa- 

 tion terms involving -j- and -j-; and then, supposing there are 

 no extraneous forces, it reduces itself to the following : 



* W~ a *\^ + ay) + af ^ ~ f W ~ 2f Tz dzTt 



J dz* dz* 



If we suppose /to have a minimum value where #=0 and y = 0, 

 the multiplier of (/> in the above equation will be a positive quan- 

 tity, and the integration of the equation will lead to logarithmic 

 functions admitting of no interpretation relative to the motion. 

 We must, therefore, suppose /to have a maximum value, which, 

 if we please, may be unity. Then we shall also have by the 

 conditions of a maximum, 



d?f d?f 1? 



dx 2 dy 2 a 2 ' 



IP being some constant as yet unknown. Again, as the present 

 investigation is independent of any arbitrary condition, the arbi- 

 trary function F ; (/) has no signification, and we may therefore 

 suppose it to be zero, or ~F(t) to be a constant. The equation 

 then becomes 



C\-ffi t k-„9 d ** + d ** 4. 2 d * • d ** 4- d ** . ^ (S\ 



°- b * a d? + -dP +2 d*~ dzTt + d? "W m ' () 



22. The integration of this equation may be expected to give 



L2 



