The Advanced Part of A Treatise on the Dynamics of a System of Rigid Bodies: Being Part II. of a Treatise on the Whole Subject, Part 2 |
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Common terms and phrases
angle angular momentum angular velocity axis of figure body centre of gravity column cone constant coordinates cos² curve deduce determinant differential coefficients differential equation direction disturbing force dt dt dy/dt earth ecliptic ellipsoid equal roots equations of motion expression fixed in space fixed point follows function geometrical equations given in Art Hence horizontal inertia initial conditions initial values instantaneous axis integral invariable line Lagrange's Lagrange's equations moments moments of inertia moving axes multipliers negative Nutation obtain oscillation P₁ particle period perpendicular plane polhode precession principal axes prove quadric radius ratio represented result right-hand side rotation sin² small quantities solution sphere steady motion straight line string Substituting suppose tangent theorem vertical vis viva viva w₁ w₂ Y₁ zero аф
Popular passages
Page 384 - Mariners 6 and 7 have been used to obtain values for the ratio of the mass of the Earth to that of the Moon which are in substantial agreement with those determined from other Mariner and Pioneer spacecraft.
Page 278 - B, a, ft are the same for all the particles, but not necessarily the same for all the trigonometrical terms defined by the different values of p. When we wish to discuss the properties of any particular A and B we write as a suffix the letter p by which they are distinguished. * This equation might also be deduced from Lagrange's general equations of motion.
Page 452 - ... always in this plane in the nearest direction to that pointed by its magnetic axis; that is, the direction of the orthogonal projection of the magnetic axis on the plane ; or the direction that the steel needle would point with its magnetic axis if placed with it in the plane, and free to turn about an axis through its centre of gravity perpendicular to the plane.
Page 208 - Problems for 1860, page 92. 7. Every particle of a sphere of radius a, which is placed on a perfectly rough sphere of radius c, is attracted to a centre of force on the surface of the fixed sphere with a force varying inversely as the square of the distance ; if it be placed at the extremity of the diameter through the centre of force and be set rotating about that diameter and then slightly displaced, determine its motion ; and show that when it leaves the fixed sphere the distance of its centre...
Page 221 - In many cases it is. impracticable to solve this equation and therefore the motion cannot be properly found. If however we only wish to ascertain whether the position of equilibrium or the steady motion about which the system is in oscillation is stable or unstable we may proceed without solving the equation. It is clear from Art. 282 that the conditions of stability are that the real roots and the real parts of the imaginary roots should all be negative. It is now proposed to investigate a method...
Page 304 - Maupertuis conceived that he could establish a priori, by theological arguments, that all mechanical changes must take place in the world so as to occasion the least possible quantity of action. In asserting this, it was proposed to measure the Action by the product of Velocity and Space ; and this measure being adopted, the mathematicians, though they did not generally assent to Maupertuis' reasonings, found that his principle expressed a remarkable and useful truth, which might be established on...
Page 76 - Routh arrives at the following results : 1°. The motion cannot be stable unless k is less than 3. 2°. The motion is stable whatever the masses may be if the law of force be expressed by any positive power of the distance or any negative power less than unity. For other powers the stability will depend on the relation between the masses. 3°. The motion is stable to a- first approximation if m')' (l+ ' Mm + Mm' + mm' \3 - k) ' where J/, m, m
Page 153 - ... to rise as long as it is kept up by the action of the air beneath. In like manner the boomerang, as long as the forward movement imparted to it by the thrower continues, will continue to rise, and the plane of rotation, instead of continuing perfectly parallel to its original position, will be slightly raised by the action of the atmosphere on the forward side. When the movement of transition ceases, the boomerang will begin to fall, and its course in falling will be by the line of least resistance,...
Page 89 - That the straight line whose equations referred to the moving principal axes are x\A wj = y/Bw^ = z/Cu.j is absolutely fixed in space may be also proved thus, if we assume the truth of equation (1) in the text. Let x, y, z be the co-ordinates of any point P in the straight line at a given distance r from the origin, then each of the equalities in the equation to the straight line is equal to rjG and is therefore constant.
Page 246 - The remarkable phenomenon discovered by Foucault, and rediscovered and extended by Kirchhoff, that a body may be at the same time a source of light, giving out rays of a definite refrangibility, and an absorbing medium, extinguishing rays of that same refrangibility which traverse it, seems readily to admit of a dynamical illustration borrowed from sound.