EXERCISES. 253 



4. In the case of motion in two dimensions for which 



udx + vdy = I 



apply (10) of Art. 10 to obtain the general equation of lines made up of 

 the same particles; and thence shew that the particles which once lie in 

 a curve of the nth order continue to lie in a curve of the wth order. 



[Stokes.] 



5. Investigate an expression for the change in an indefinitely short 

 time in the mass of fluid contained within a spherical surface of small 

 radius, 



Prove that the momentum of the mass in the direction of the axis of 

 x is greater than it would be if the whole were moving with the velocity 

 at the centre by 



1 Ma 2 {dp du dp du dp du fd~n dru &amp;lt;Fu\\ 

 ~ ++ ~ + + + 



[H. M. Taylor, Math. Trip., 1876.] 



6. A cistern discharges water into the atmosphere through a verti 

 cal pipe of uniform section. Shew that air would be sucked in through 

 a small hole in the upper part of the pipe, and explain how this result 

 is consistent with an atmospheric pressure in the cistern. 



[Lord Rayleigh, Math. Trip., 1876.] 



7. Let a spherical portion of an infinite quiescent liquid be separated 

 from the liquid round it by an infinitely thin flexible membrane, and let 

 this membrane be suddenly set in motion, every part of it in the direc 

 tion of the radius and with velocity equal to S t , a harmonic function of 

 position on the surface. Find the velocity produced at .any external or 

 internal point of the liquid. [Thomson.] 



8. Prove that the energy of the irrotational motion of a liquid in a 

 given region is less than that of any other continuous motion consistent 

 with the same motion of the boundary. [Thomson.] 



9. Prove that if the force-potential V satisfy the relation v/T^ 

 the pressure cannot, in irrotational motion, be a minimum at any point 

 in the interior of an incompressible fluid. 



10. In the irrotational motion of a fluid in two dimensions prove 

 that if the velocity be everywhere the same in magnitude it is so in 

 direction. [Math. Trip., 1873.] 



