## A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion |

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arbitrary constants asymptotes axes axis biquadratic centrifugal forces change sign changes of sign Chap Chapter co-ordinates column conditions of stability convenient derived equation derived P curve determinant determinantal equation differential coefficients differential equation dimensions dynamical system eliminate equa equal roots equation f(x equations of motion equations to find force function ftft fundamental term geometrical equations greatest common measure Hamiltonian harmonic oscillation Hence highest powers homogeneous function imaginary roots integrals kinetic energy Lagrange's equations method minimum motion is stable necessary and sufficient number of factors obtain omitting plane PoP3 position of equilibrium potential energy proper number Q curve quadric quantities radical point real roots roots equal rows satisfy second minors separate the roots shown steady motion straight line Sturm's theorem Substituting terms which contain theorem tion undisturbed motion unstable value of m2 variations of sign velocity vertical

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Página v - PRIZE," be awarded every two years to the author of the best Essay on some subject of Pure Mathematics, Astronomy, or other branch of Natural Philosophy. 2. That the...

Página vi - ... and the recently discovered dark ring may be most satisfactorily explained ; and to indicate any cause to which a change of form, such as is supposed, from a comparison of modern with the earliest observations, to have taken place, may be attributed. " The essay must be sent in to the Vice-Chancellor on or before the 16th of October, 1856, privately ; each is to have some motto prefixed, and to be accompanied by a paper sealed up, with the same motto outside, which paper is to inclose another,...

Página v - Prize be open to the competition of all persons who have at any time been admitted to a Degree in this University. 4. That the successful Candidate receive two years...

Página v - The Criterion of Dynamical Stability. To illustrate the meaning of the question imagine a particle to slide down inside a smooth inclined cylinder along the lowest generating line, or to slide down outside along the highest generating line. In the former case a slight derangement of the motion would merely cause the particle to oscillate about the generating line, while in the latter case the particle would depart from the generating line altogether. The motion in the former case would be, in the...

Página 12 - These criteria were derived by a new transformation, namely the one which yields an equation whose roots are the squares of the differences of the roots of the given...

Página v - The Examiners give notice that the following is the subject of the Prize to be adjudged in 1877 : The Criterion of Dynamical Stability. To illustrate the meaning of the question imagine a particle to slide down inside a smooth inclined cylinder along the lowest generating line, or to slide down outside along the highest generating line. In the former case a slight derangement of the motion would merely cause the particle to oscillate about the generating line, while in the latter case the particle...

Página v - The University having accepted a Fund raised by several members of St John's College for the purpose of founding a Prize to be called the Adams Prize, for the best essay on some subject of Pure Mathematics, Astronomy or other branch of Natural Philosophy, the Prize to be given once in two years, and to be open to the competition of all persons who have at any time been admitted to a degree in this University...

Página 43 - BAB' makes with a fixed vertical plane. Let m be the mass of either sphere and rod; k the radius of gyration about an axis through G perpendicular to the rod, and k' that about the rod, let I = A G.

Página 61 - Clairaut's form, of which the complete primitive is and the singular solution The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two. 3. A more general class of problems is that in which it is required to determine the curves in which some one of...