TRANSACTIONS OF THE SECTIONS. 11 



some elegant theorems of kinematics discovered by M. Chasles. Demonstrations 

 of some of the theorems here enunciated ■will be found in two papers written by 

 the author. 



"On the small oscillations of a Particle on a Surface under the action of any 

 Forces," Quarterly Journal of Mathematics, No. 39, 1869. 



" On the small oscillations of a Rigid Body about a Fixed Point under the action 

 of finy Forces, and more particularly when gravity is the only force acting," Trans- 

 actions of the Royal Irish Academy, vol. xxiv. Science, part xvi. 



II, A Particle. 



1. There are in general three lines called normal lines, such that whatever be the 

 small osciUatious of a particle, free in space, the movement is compounded of simple 

 harmonic vibrations along the normal lines. 



2. When the forces have a potential, a constant small quantity of energy would 

 di'aw the particle along any radius vector from its position of rest to the surface of 

 a certain ellipsoid ; the normal lines are in the principal directions of this ellipsoid, 

 and the lengths of the isochronous simple pendulums are proportional to the squares 

 of its principal axes. 



3. When the particle is constrained to a surface, the motion is compounded of 

 vibrations in two directions on the surface, and when the forces have a potential, 

 the tangent lines to these directions are at right angles. 



III. A Free Rigid Body, 



4. A free rigid body may receive any displacement by being screwed along an 

 axis in space, the distance it travels along the axis when turned through the unit 

 of angle Ibeing termed the pitch of the screw. 



5. The movement of a free rigid body, when making small oscillations, is com- 

 pounded of six normal movements, each consisting of a to-and-fro vibration about 

 a tiormid screw, the position, pitch, and period of which depend upon the forces. ' 



6. Whatever be the initial motion of the body, supposed small, it may be dis- 

 tributed imiquely among the six nonnal screws, and thus the entire motion is de- 

 termined, 



rV. A Constrained liir/id Body. 



7. If a rigid body have h degrees of freedom, its small oscillations are compounded 

 of vibrations about k normal screws. 



8. A body capable of turning around a fixed axis and sliding along it, has two 

 degrees of freedom ; its motion is compounded of that about two normal screws 

 whose pitch is different, but both of which lie in the fixed axis. 



9. A body three points of which are limited to a plane, has three degi'ees of 

 freedom ; its motion is compounded of vibrations about three nonnal screws 

 whose pitch is zero, and whose directions are perpendicular to the plane. 



10. A body rotating about a fixed point has three degrees of freedom; its 

 motion is compounded of \'ibrations about three normal screws whose pitch is zero, 

 and whose directions pass through the point. 



N.B. — The screws in this case may be conveniently called the normal axes. 



V. A Rigid Body rotating about a Point, the Forces having a Potential. 



11. The body may be moved from one position to any other position by rotation 

 about a certain axis, passing through the point through a certain angle; this 

 axis and angle are called the axis of displacement and the angle of displacement 

 respectively. 



12. On an axis through the point, take a radius vector proportional to the small 

 angular velocity, which a small quantity of energy would be able to communicate to 

 the body about the axis. The quantity of energy being constant, the locus of this 

 point on different axes is the momenta! ellipsoid. 



13. On an axis through the point, take a radius vector proportional to the small 

 angle through which a small quantity of energy would be able to rotate the body 

 about the axis from its position of equilibrium against the forces. The quantity of 

 energy being constant, the locus of this point on different axes may be called the 

 ellipsoid of equal energy. 



14. The three common conjugate diameters of the momental ellipsoid and the 



