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Seite xx - ... coinciding double points, and the polar of this point with respect to the imaginary circle on the plane at infinity cuts that circle in the two other double points. The displacement of the rigid body can thus be produced either by rotating the body around S or by translating the body parallel to S, or by a combination of such movements. We are therefore led to the fundamental theorem discovered by Chasles. Any given displacement of a rigid body can be effected by a rotation about an axis combined...
Seite 191 - ... ellipsoid cuts the momental ellipsoid and the cylinder. These three lines are the three harmonic axes. As to that vertical axis which appears to be one of the harmonic axes, the time of vibration about it would be infinite. The three harmonic screws which are usually found in the small oscillations of a body with freedom of the third order are therefore reduced in the present case to two, and we have the following theorem : — A rigid body which is free to rotate about a fixed point is at rest...
Seite 1 - ... termed the instantaneous screw. 7. Definition of the word Wrench. It has been explained in the Introduction that a system of forces acting upon a rigid body may be generally expressed by a certain force and a couple whose plane is perpendicular to the force. We now employ the word wrench, to denote a force and a couple in a plane perpendicular to the force. The quotient obtained by dividing the moment of the couple by the force is a linear magnitude. Everything, therefore, which could be specified...
Seite 57 - XR tan 6, where sin 0 is the eccentricity of the ellipse. Also 6 is the angle between the normal to the plane and the nodal axis. Fig. 17. Let two circles be described, one with the major axis of the ellipse as diameter and the other with the line joining the two foci as a diameter. Let Xt be the point in which the ordinate through X meets the first circle and X« be the point in which a ray drawn from X , to the centre meets the second circle.
Seite 24 - This plane can cut the cylindroid in a conic section only, for the line LM and the conic will then Fig. 4. make up the curve of the third degree, in which the plane must intersect the surface. Also since the entire cylindroid (or at least its curved portion) is included between two parallel planes (§ 17), it follows that this conic must be an ellipse. We shall now prove that this ellipse is the locus of the feet of the perpendiculars let fall from 0 on the generators of the cylindroid. Draw in the...
Seite 166 - On the small oscillations of a rigid body about a fixed point under the action of any forces, and more particularly when gravity is the only force acting.
Seite 448 - A, from y to z and y to x be denoted by B, and from z to x and z to y be denoted by C. Then, sin A _ sin B _ sin C sin a ~ sin 6 ~ sin c ' cos a = cos b cos c + sin b sin c...
Seite 3 - For example, pa denotes the pitch of a and is an ordinary algebraical quantity. 3. Definition of the word Twist. We have next to define the use to be made of the word twist. A body is said to receive a twist about a screw when it is rotated uniformly about the screw, while it is translated uniformly parallel to the screw, through a distance equal to the product of the pitch and the circular measure of the angle of rotation. 4. A Geometrical Investigation. We can now demonstrate...
Seite 306 - Let now ft and f be two other screws (not reciprocal) : we may consider the question as to whether a rigid body can be designed and placed so that a shall be the instantaneous screw corresponding to rj as an impulsive screw, while ft bears the same relation to f.
Seite 429 - Proceedings of the London Mathematical Society, Vol. iv. 381—395 (1873). See also "On the Theory of Screws in a Space of Constant Positive Curvature," Mathematical Papers, p. 402 (1876). Clifford's Theory was much extended by the labours of Buchheim and others ; see the Bibliographical notes. t We are fortunately now able to refer English readers to a Treatise in which the...