The same formula will apply to any other internal particle, and it has been shewn in Art. (243), that for an external particle NdS= 0. Hence, adding together all the results, and taking N now to refer to the attraction of all the particles, both internal and external, we get [NdS=- 4πM. which proves the proposition.

dV But N = dn

245. For the researches of M. Chasles on the attraction of ellipsoids, we refer to Duhamel's Cours de Mécanique, or to the original memoirs in the Journal de l'Ecole Polytechnique, tom. XV., and the Mémoires....... Savans Etrangers, tom. IX. In the original memoirs will be found copious references to preceding writers on the subject.

On the general theory of attractions, the student may consult a memoir by Gauss, translated in Taylor's Scientific Memoirs, vol. III., and in Liouville's Journal de Mathéma tiques, tom. VII.; and also a memoir by M. Chasles in the Connaissance des Temps pour l'année 1845.

Some further references will be seen in the article by Professor Stokes already cited.

For the application to the theory of electricity, we refer to a series of articles by Professor Thomson in different volumes of the Cambridge and Dublin Mathematical Journal, see vol. I. p. 94, and vol. III. p. 140.

EXAMPLES.

In the following Problems the ordinary law is to be assumed, unless the contrary be stated.

1. Determine the attraction of a right prism having a square base on a particle on the production of its axis.

2. A solid is generated by the revolution of a sector of a circle about one of its bounding radii; find the attraction on a particle at the centre.

3. If be the angle between any two lines in space, the whole attraction in the direction of the shortest line 2π between them is

sin; the mutual attraction of two units of length collected in centres, and separated by the unit of distance, being considered equal to unity.

4. The rim of a hemispherical bowl consists of matter repelling with a force varying directly as the distance; shew that a particle will rest when placed anywhere on the concave surface.

5. A tube in the form of a parabola is placed with its axis vertical and vertex downwards; a heavy particle is placed in the tube, and a repulsive force acts along the ordinate upon the particle: find the law of force that it may sustain the particle in any position.

6. Eight centres of force, resident in the corners of a cube, attract, according to the same law and with the same absolute intensity, a particle placed very near the centre of the cube; shew that their resultant attraction passes through the centre of the cube, unless the law of force be that of the inverse square of the distance.

7. A portion of a cylinder of uniform density is bounded by a spherical surface, the radius of which is greater than that of the cylinder, and the centre coincides with the middle point of the base; find the attraction on a particle at this point.

8. The attraction of a uniform rod of indefinite length on an external particle varies as (distance) of the point from the rod. Prove this, and supposing the asymptotes of an hyperbola to consist of such material, shew that a particle will be in equilibrium at any point of the hyperbola, and that the pressure on the curve at any point is proportional to the length of the tangent intercepted by the asymptotes.

9. An elliptic board attracts an internal particle (x, y) with a force varying inversely as the distance; shew that if X, Y be the whole attractions parallel to the axes,

10. If a portion of a thin spherical shell, whose projections upon the three coordinate planes through the centre are A, B, C, attract a particle at the centre with a force varying as any function of the distance, shew that the particle will begin to move in the direction of a straight line whose equations are

11. The particles of a thin hemispherical shell attract with a force = μ (distance), and those of a right conical shell repel with a force = (distance). The rims of their bases coincide, and their vertices are turned in opposite directions,

shew that a particle will rest in the common axis produced at a distance from the vertex of the sphere = length of the axis of the cone, the vertical angle of the cone being 2 tan1.

12. If a straight line be the attracting body, shew that the lines of force are hyperbolas and the surfaces of equilibrium spheroids. (Cambridge and Dublin Mathematical Journal, vol. III. p. 94.)

13. From the proposition established in Art. (244), deduce that established in Art. (239). (Cambridge and Dublin Mathematical Journal, vol. v. p. 215.)

14. The attraction of an ellipsoidal shell on an external particle is in the direction of the axis of the enveloping cone, whose vertex is the given particle. (Crelle's Journal, vol. XII. p. 141.)

CHAPTER XIV.

VIRTUAL VELOCITIES.

246. WE proceed to establish a general theorem respecting the equilibrium of a body or system of bodies, called the Principle of Virtual Velocities.

DEF. When a system of particles is in equilibrium, and we suppose each of them placed in a position indefinitely near that which it really occupies, without disturbing the connexion of the parts of the system with each other, the line which joins the first position of a particle with the second is called the virtual velocity of that particle.

The term velocity is used because we may conceive all the displacements to be made in the same indefinitely small time, and then the spaces described are proportional to the velocities. The word virtual is used to intimate that the displacements are not really made, but only supposed. We retain the established phraseology, but it is evident from these explanations that the words virtual velocity might be conveniently replaced by hypothetical displacement.

By the words, without disturbing the connexion of the parts of the system with each other, we mean, that any rigid body which exists in the system is supposed to remain of invariable form, and that any rods or strings which connect different parts of the system are to remain unbroken. This, at least,