LIST OF MEMOIRS AND TREATISES.

MEMOIRS.

EULER, L. Principes généraux du mouvement des fluides. Hist. de l'Acad. de Berlin. 1755.

De principiis motus fluidorum. Novi Comm. Acad. Petrop. t. 14, p. 1.

1759.

LAGRANGE, J. L. Mémoire sur la théorie du mouvement des fluides. Nouv. Mem. de l'Acad. de Berlin. 1781.

POISSON, S. D. Sur les équations générales de l'équilibre et du mouvement des corps solides élastiques et des fluides. Journ. de l'Ecole Polyt. t. 13. 1819.

GREEN, G. Researches on the vibrations of pendulums in fluid media. Trans. R. S. Edin. Vol. 13. 1833.

On the motion of waves in a variable canal of small depth and width. Camb. Trans., vol. 6, 1837.

Note on the motion of waves in canals. Camb. Trans., vol. 7. 1839.

STOKES, G. G. On the steady motion of incompressible fluids. Camb.

Encyc. Metrop., vol. 5. 1845.

SCOTT RUSSELL, J. Report on waves. B. A. Reports. 1844.

On some cases of fluid motion.

STOKES, G. G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Camb. Trans., vol. 8. 1845.

Report on recent researches in hydrodynamics. B. A. Reports.
1846.

Supplement to a memoir on some cases of fluid motion. Camb.
Trans., vol. 8. 1846.

On the theory of oscillatory waves. Camb. Trans., vol. 8.
1847.

THOMSON, W. Notes on hydrodynamics. Camb. and Dub. Math. J., vol. 4. 1849.

STOKES, G. G. On waves.

Camb. and Dub. Math. J., vol. 4. 1849.

On the effect of the internal friction of fluids on the motion of pendulums. Camb. Trans., vol. 9. 1851.

CLEBSCH, A. Ueber die Bewegung eines Ellipsoids in einer tropfbaren Flüssigkeit. Crelle, t. 52. 1856; and t. 53. 1857.

HELMHOLTZ, H. Ueber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen. Crelle, t. 55. 1858.

RIEMANN, B. Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gött. Abh., t. 8. 1860.

DIRICHLET, P. LE-J. Untersuchungen über ein Problem der Hydrodynamik. Gött. Abh., t. 8. 1860.

EARNSHAW, S. On the mathematical theory of sound. Phil. Trans. 1860.

HELMHOLTZ and PIOTROWSKI. Ueber Reibung tropfbarer Flüssigkeiten. Wien. Sitz. B. t. 40. 1860.

MEYER, O. E. Ueber die Reibung der Flüssigkeiten. Crelle, t. 59. 1861.

RIEMANN, B. Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoids. Gött. Abh., t. 9. 1861. RANKINE, W. J. M. On the exact form of waves near the surface of deep water. Phil. Trans. 1863.

STEFAN, M. J. Ueber die Bewegung flüssiger Körper. Wien. Sitz. B.

t. 46. 1863.

RANKINE, W. J. M. On plane water-lines in two dimensions. Phil. Trans. 1864.

Summary of the properties of certain stream-lines. Phil. Mag. 1864.

MAXWELL, J. C. On the viscosity or internal friction of air and other gases. Phil. Trans. 1866.

THOMSON, W.

On vortex atoms. Phil. Mag. 1867. RANKINE, W. J. M. On waves in liquids.

Proc. R. S. 1868.

HELMHOLTZ, H. Ueber discontinuirliche Flüssigkeitsbewegungen.

Berl. Monatsb.

1868.

WEBER, H. Ueber eine Transformation der hydrodynamischen Gleichungen. Crelle, t. 68. 1868.

RANKINE, W. J. M. On the thermodynamic theory of waves of finite

longitudinal disturbance.

THOMSON, W. On vortex motion.

Phil. Trans.

1870.

Trans. R. S. Edin., vol. 25. 1869.

KIRCHHOFF, G. Zur Theorie freier Flüssigkeitsstrahlen. Crelle, t. 70.

1869.

Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit.
Crelle, t. 71.

1870.

Ueber die Kräfte, welche zwei unendlich dünne, starre Ringe in einer Flüssigkeit scheinbar auf einander ausüben können. Crelle, t. 71.

1870.

MAXWELL, J. C. On the displacement in a case of fluid motion. Proc. Lond. Math. Soc., vol. 3. 1870.

MEYER, O. E. Ueber die pendelnde Bewegung einer Kugel unter dem Einflusse der inneren Reibung des umgebenden Mediums. Crelle, t. 73.

1871.

RANKINE, W. J. M. On the mathematical theory of stream-lines, especially those with four foci and upwards. Phil. Trans. 1871. THOMSON, W. Hydrokinetic solutions and observations. Phil. Mag. 1871. On the ultra-mundane corpuscles of Le Sage, also on the motion of rigid solids in a liquid circulating irrotationally through perforations in them or in a fixed solid. Phil. Mag. 1873. RAYLEIGH, LORD. On waves. Phil. Mag. 1876.

On the resistance of fluids. Phil. Mag. 1877.

Notes on hydrodynamics. Phil. Mag. 1877.

GREENHILL, A. G. Plane vortex motion. Quart. J. of Math., vol. 15.

1877.

TREATISES.

LAGRANGE, J. L. Mécanique analytique. Ed. 1815, Part 2, Sect. 10,

11, 12.

POISSON, S. D. Traité de mécanique. 2me éd. Paris, 1833. T. 2, liv. 6.

THOMSON, W. and TAIT, P. G. Treatise on natural philosophy. Oxford, 1867. §§ 331-336. New edition: Cambridge, 1879. §§ 320-330.

Braun

RIEMANN, B. Partielle Differentialgleichungen und deren Anwendung auf physikalische Fragen. Hrsg. v. Hattendorff. schweig, 1869. §§ 98-109.

RÉSAL, H. Traité de mécanique générale. Paris, 1874.

13, 14.

KIRCHHOFF, G.

T. 2, cc.

Vorlesungen über mathematische Physik. Leipzig,

1874-6. Cc. 15-26.

BESANT, W. H. Treatise on Hydromechanics. 3rd edition, Cambridge,

1877. Cc. 10-15.

EXERCISES.

1. If the motion of a fluid in two dimensions be referred to polar co-ordinates r, 0, and if u, v denote the component velocities along and perpendicular to the radius vector, the component accelerations in the same directions are

where u, v, w are given functions of x, y, z, t, satisfying the condition

x, y, z be expressed as functions of t, and the arbitrary constants a, b, c,

3. If F(x, y, z, t)=0 be the equation of a moving surface, the velocity of the surface normal to itself at any point is

4.

In the case of motion in two dimensions for which

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 nth 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

6.

A cistern discharges water into the atmosphere through a vertical 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 direction of the radius and with velocity equal to S, 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 = 0 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.]