Prof. Challis on Hydrodynamics. 215 



tions of the motion may be satisfied. Let us first suppose that 

 where r is indefinitely great, and consequently t-¥l an d y D0 th 

 vanish, the pressure p has the constant value II. Then 



which shows that if the velocity either be a maximum, or be con- 

 stant at a given distance from the centre, p is less than II, and 

 the difference depends only on the square of the velocity. Again, 

 suppose the fluid to be put in motion by being continuously 

 impressed at the distance c with a given velocity equal to m sin lit. 

 Then, by what is proved above, the velocity at any distance r is 



7YIC 



~ sin ht, and f(t) = mc* sin ht. Hence f'(t) = mcVi cos ht, and 



t-t mcVi _ - m 2 c 4 . , 



p — 11 = cos ht — — r sm 2 ht. 



r 2r 4 



Thus the motion and pressure are completely determined. 

 At the distance c, when t=0, pr=Tl-\-mch. This result shows 

 that the fluid, although of infinite extent, may be started at a 

 finite distance from the centre by a pressure exceeding II by a 



finite amount. When ht = -, and consequently the velocity is a 



mC 77b c 



maximum, V= — T , and p = Ii— — Ti showing that the pres- 

 sure is greater as r is greater. This is also true if the velocity 

 be constant at a given distance, and evidently explains why in 

 that case the velocity of a given particle diminishes as its dis- 



mc 

 tance from the centre increases. When V= — «-, the total mo- 



mentum of the fluid is 47rmc 2 (r— c), which is an infinite quan- 

 tity, r being infinite. Consequently an infinite amount of mo- 

 mentum may be generated in a finite interval. This peculiarity 

 of incompressible fluid in motion is analogous to what is called 

 " the hydrostatic paradox." The solution of the problem of the 

 central motion of an incompressible fluid presents no difficulty. 

 Let us now treat in a similar manner the central motion of a 

 compressible fluid. It will suffice for the present purpose to 

 restrict the reasoning to the first approximation, for which we 

 have the known equations 



d*.rcf>_ 2 d*r$_ # 



