EXERCISES. 255 



17. A large closed vessel of incompressible liquid with a motionless 

 solid globe immersed in it at a distance from the nearest part of the 

 outer boundary very great in comparison with the radius of the globe, 

 is suddenly set in motion with a given translational velocity V. Find 

 the instantaneous motion of the liquid and of the globe. Prove that 

 according as the mean density of the globe is greater or less than the 

 density of the liquid, its velocity is less or greater than V. (Assume 

 the centre of inertia of the globe to be in its centre of figure.) 



[Thomson.] 



18. Prove that under the circumstances of Arts. 107, 108 the 

 linear momentum of the whole matter contained within a spherical 

 surface enclosing the whole of the moving solids is equal to two-thirds 

 of the impulse ; also that the linear momentum of the matter contained 

 within a spherical surface not enclosing any of the solids is the same as 

 if this matter were rigid and moving with the velocity at the centre. 



[Thomson.] 



1 9. Verify the former of the statements in the preceding exercise in 

 the case of the sphere (Art. 105). 



20. A sphere is in motion in a mass of liquid enclosed in a fixed 

 spherical case. Find the motion of the liquid at the instant when the 

 centre of the sphere coincides with that of the case. [Stokes. 1 



21. Also workout, by the method of Art. 125, an approximate solu 

 tion for the case in which the distance between the centres is small 

 compared with the radius of the containing vessel. 



22. Integrate the equations of motion of a solid through a liquid in 

 the case of Art. 116 (g). [Thomson.] 



23. Work out the problem of Art. 125, supposing the spheres to be 

 moving perpendicularly to the line of their centres. 



24. Prove (by the method of images) that a straight vertical vortex- 

 filament moving in water bounded by two vertical walls at right angles 

 will describe the Cotes s spiral r sin 20= a in exactly the same manner 

 as a particle under the action of a repulsion varying inversely as the 

 cube of the distance. 



Prove also by the same method that if the vertical planes be inclined 

 at an angle -, an exact submultiple of two right angles, the vortex- 



77* 



filament will describe similarly the Cotes s spiral r sin nO = a. 



[Greenhill.] 



