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dynamical

dynamical Sentence Examples

  • Devoting himself next to optics, he produced memoirs which entitle him to a high place among the early, searchers after a true dynamical theory of light.

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  • Though we may allow that the results obtained by Rumford and Davy demonstrate satisfactorily that heat is in some way due to motion, yet they do not tell us to what particular dynamical quantity heat corresponds.

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  • The dynamical equilibrium between rhombic, liquid and monosymmetric sulphur has been worked out by H.

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  • Therefore, according to Kekule, the double linkages are in a state of continual oscillation, and if his dynamical notion of valency, or a similar hypothesis, be correct, then the difference between the 1.2 and 1.6 di-derivatives rests on the insufficiency of his formula, which represents the configuration during one set of oscillations only.

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  • A dynamical theory due to E.

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  • The heading "Measurement of Dynamical Quantities" includes the topics units, measurements, and the constant of gravitation.

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  • Mechanics (including dynamical astronomy) is that subject among those traditionally classed as "applied" which has been most completely transfused by mathematics - that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.

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  • The dynamical equivalent of the calorie is 4.18 X Io 7 ergs or C.G.S.

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  • Thomson (Applications of Dynamics to Physics and Chemistry, 47) that on dynamical principles there must be a reciprocal relation between the changes of dimensions produced by magnetization and the changes of magnetization attending mechanical strain.

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  • Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor theorems of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.

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  • Although the matter can be fully treated only upon the basis of a dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens's principle to the calculation of the effects produced by opaque screens of limited extent.

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  • Dynamical Theory of Diffraction.

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  • This question was treated by Stokes in his " Dynamical Theory of Diffraction " (Camb.

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  • Cohnheim (1839-1884) and of Iliya Metchnikoff on the dynamical side of his- Fevers tology.

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  • Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.

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  • In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?

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  • Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

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  • When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express the fact that x, x, xs.

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  • the precedin investigation, the liquid stops dead when the body is brought to rest and when the body is in motion the surrounding liquid moves in uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on it surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations give above.

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  • 00 ab2dX °0 ab2dX (22) L (I JS)' (18) (t9) This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.

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  • Social physics will consist of the conditions and relations of the facts of society, and will have two departments, - one, statical, containing the laws of order; the other dynamical, containing the laws of progress.

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  • In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

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  • A number of molecules moving in obedience to dynamical laws will pass through a series of configurations which can be theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known.

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  • We begin with a general dynamical theorem, whose special application, when the dynamical system is identified with a gas, will appear later.

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  • qr,, be the generalized coordinates of any dynamical system, and let pi, P2,

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  • are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, v, w lie within ranges such that (i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds: (ii) E2mu2 E2mv 2 E2mw2 ?2aie12 s S s s s s - s E s' S' s' - - s' ' See Jeans, Dynamical Theory of Gases (1904), ch.

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  • The planetary theory and other particular dynamical problems likewise occupied his attention from time to time.

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  • Lamb, The Dynamical Theory of Sound (1910), is intended as a stepping-stone to the study of-the writings of Helmholtz and Rayleigh.

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  • These antinomies are four - two mathematical, two dynamical - connected with (I) the limitation of the universe in respect of space and time, (2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist), (3) the problem of freedom in relation to universal causality, (4) the existence of a universal being - about each of which pure reason contradicts the empirical, as thesis and antithesis.

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  • It has been repeatedly claimed for Mayer that he calculated the value of the dynamical equivalent of heat, indirectly, no doubt, but in a manner altogether free from error, and with a result according almost exactly with that obtained by J.

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  • The wider view, according to which the hypothesis of direct transmission of physical influences expresses only part of the facts, is that all space is filled with physical activity, and that while an influence is passing across from a body, A, to another body, B, there is some dynamical process in action in the intervening region, though it appears to the senses to be mere empty space.

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  • MacCullagh's hands the correct equations were derived from a single energy formula by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming.

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  • In this place it must suffice to indicate the gist of the more recent developments of the electro-optical theory, which involve the dynamical verification of Fresnel's hypothesis regarding optical convection and the other relations above described.

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  • The dynamical series of stages in nature, the forms in which the ideal structure of nature is realized, are matter, as the equilibrium of the fundamental expansive and contractive forces; light, with its subordinate processes - magnetism, electricity, and chemical action; organism, with its component phases of reproduction, irritability and sensibility.'

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  • Just as nature exhibits to us the series of dynamical stages of processes by which spirit struggles towards consciousness of itself, so the world of intelligence and practice, the world of mind, exhibits the series of stages through which self-consciousness with its inevitable oppositions and reconciliations develops in its ideal form.

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  • Lord Rayleigh,' who has also investigated vibrating systems giving series of lines approaching a definite limit of " root," remarks that by dynamical reasoning we are always led to equations giving the square of the period and not the period, while in the equation representing spectral series the simplest results are obtained for the first power of the period.

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  • The discovery of the magnetic rotation of the plane of polarized light, though it did not lead to such important practical applications as some of Faraday's earlier discoveries, has been of the highest value to science, as furnishing complete dynamical evidence that wherever magnetic force exists there is matter, small portions of which are rotating about axes parallel to the direction of that force.

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  • A definition of the measurement dependent on dynamical theory has Time.

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  • Nevertheless, the theoretical possibility, and its realization in many cases, has brought considerations to the front which have recently become of predominant interest; consequently the possible transformations of isomers and polymers will be considered later under the denomination of reversible or dynamical isomerisms.

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  • Th~ axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions.

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  • But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another.

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  • In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios.

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  • For the determination of the motion it has only, been necessary to use one of the dynamical equations.

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  • qX +Pu = N.j These equations are applicable to any dynamical system whatever.

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  • When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are ~rn+qi=X, ~pl+rE=Y, ~f_qE+Pi,=Z, (II)

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  • As in the case of (2), the equations are applicable to any dynamical system whatever.

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  • These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated.

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  • Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical equations.

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  • Arago's previous investigations in the new science of electromagnetism, and crowned that labour by the announcement of his great discovery of the dynamical action between conductors conveying the electric currents.

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  • for 1865, entitled " A Dynamical Theory of the Electromagnetic Field."

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  • This fruitful conception, however, Bacon does not work out; and though he uses the word cause, and identifies form with formal cause, yet it is perfectly apparent that the modern notions of cause as dynamical, and of nature as in a process of flow or development, are foreign to him, and that in his view of the ultimate problem of science, cause meant causa immanens, or underlying substance, effects were not consequents but manifestations, and nature was regarded in a purely statical aspect.

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  • Invested, as society grows more complex, with a sanctity increasingly superior to that of the layman, the priest-king becomes the representative of the community as repository of its luck, whilst, as controller of all sacred forces that bear thereon, he is, as Dr Frazer puts it, " dynamical centre of the universe" (The Golden Bough (2nd ed.), i.

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  • When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x for the vapour, and po and Xo those for the liquid, x' - xo=JL - p(I/p' - I/pc),.

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  • (21) where J is the dynamical equivalent of heat, L is the latent heat of unit of mass of the vapour, and p is the pressure.

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  • The Detachment Of The Drop Is A Dynamical Effect, And It Is Influenced By Collateral Circumstances.

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  • But Although A Complete Solution Of The Dynamical Problem Is Impracticable, Much Interesting Information May Be Obtained From The Principle Of Dynamical Similarity.

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  • Dynamical Similarity Requires That T/Gva 2 Be Constant; Or, If G Be Supposed To Be So, That A' Varies As T/V.

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  • The coefficient (q) of the time in the exponential term (e at) may be considered to measure the degree of dynamical instability; its reciprocal 1 /q is the time in which the disturbance is multiplied in the ratio I: e.

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  • But they contain what is far more valuable still, the greatest addition which dynamical science had received since the grand strides made by Sir Isaac Newton and Joseph Louis Lagrange.

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  • Thus we feel it in its motion as wind, and observe the dynamical effects of this motion in the quiver of the leaf or the motion of a sailing ship. It offers resistance to the passage of bodies through it, destroying their motion and transforming their energy - as is betrayed to our hearing in the whiz of the rifle bullet, to our sight in the flash of the meteor.

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  • The practically obvious distinction between solids and fluids may be stated in dynamical language thus: - solids can sustain a longitudinal pressure without being supported by a lateral pressure; fluids cannot.

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  • The true foundations of a mechanical theory of the heavens were laid by Kepler's discoveries, and by Galileo's dynamical demonstrations; its construction was facilitated by the development of mathematical methods.

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  • The analytic method sought to express the moon's motion by integrating the differential equations of the dynamical theory.

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  • This was supplied by Fresnel, who, starting from a mechanical hypothesis, showed by ingenious but not strictly dynamical reasoning that if the incident stream have unit amplitude, that of the reflected stream will be - sin (i - r) /sin (i -{- r) or tan (i - r) /tan (i -{- r), according as the incident light is polarized in or perpendicularly to the plane of incidence i, r, being the angles of incidence and refraction connected by the formula sin i =,u sin r.

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  • The interdependence of motion and force was not indeed formulated into definite laws by Galileo, but his writings on dynamics are everywhere suggestive of those laws, and his solutions of dynamical problems involve their recognition.

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  • At first sight this period seemed to be inconsistent with dynamical theory.

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  • These changes arise from two causes - the one statical, the other dynamical.

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  • The dynamical causes are atmospheric and oceanic currents.

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  • Joule's views of the nature of heat strongly influenced Thomson's mind, with the result that in 1848 Thomson proposed his absolute scale of temperature, which is independent of the properties of any particular thermometric substance, and in 1851 he presented to the Royal Society of Edinburgh a paper on the dynamical theory of heat, which reconciled the work of N.

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  • Mayer and Joule, and placed the dynamical theory of heat and the fundamental principle of the conservation of energy in a position to command universal acceptance.

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  • According to his usual plan, Kant arranges these principles in conformity with the table of the categories, dividing the four classes, however, into two main groups, the mathematical and the dynamical.

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  • The mathematical principles are the abstract expression of the necessary mode in which data of sense are determined by the category in the form of intuitions or representations of objects; the dynamical are the abstract expression of the modes in which the existence of objects of intuition is determined.

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  • express determinations of the objects themselves; the dynamical are regulative, i.e.

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  • Under the dynamical principles, the general modes in which the existence of objects are determined, fall the analogies of experience, or general rules according to which the existence of objects in relation to one another can be determined, and the postulates of experience, the general rules according to which the existence of objects for us or our own subjective existence can be determined.

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  • Finally, the dynamical aspects of Bose condensation are further investigated by studying the evolution of excited atoms.

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  • The algorithm can also include dynamical bosons, again with any number of flavors.

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  • differentiable dynamical systems and ergodic theory (theoretical and computational aspects ).

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  • The R2 algorithm is specifically for simulating 2+1 flavors of dynamical staggered fermions of different masses.

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  • In this dynamical process the time delay between infrasonic pulses should reflect the gas nucleation interval of 1-2 s of basaltic magma.

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  • nucleon structure functions on dynamical configurations.

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  • This work involves studies of the Hybrid Monte Carlo algorithm for generating dynamical quarks.

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  • Devoting himself next to optics, he produced memoirs which entitle him to a high place among the early, searchers after a true dynamical theory of light.

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  • In 1832, by the aid of a particular hypothesis as to the constitution of the ether, he reached by a rigorous dynamical calculation results agreeing with those obtained by A.

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  • Though we may allow that the results obtained by Rumford and Davy demonstrate satisfactorily that heat is in some way due to motion, yet they do not tell us to what particular dynamical quantity heat corresponds.

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  • Dewar led him to see the true dynamical explanation of the Crookes radiometer in the largeness of the free path of the molecule of the highly rarefied air.

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  • The dynamical equilibrium between rhombic, liquid and monosymmetric sulphur has been worked out by H.

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  • Therefore, according to Kekule, the double linkages are in a state of continual oscillation, and if his dynamical notion of valency, or a similar hypothesis, be correct, then the difference between the 1.2 and 1.6 di-derivatives rests on the insufficiency of his formula, which represents the configuration during one set of oscillations only.

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  • A dynamical theory due to E.

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  • The heading "Measurement of Dynamical Quantities" includes the topics units, measurements, and the constant of gravitation.

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  • Mechanics (including dynamical astronomy) is that subject among those traditionally classed as "applied" which has been most completely transfused by mathematics - that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.

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  • The dynamical equivalent of the calorie is 4.18 X Io 7 ergs or C.G.S.

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  • Thomson (Applications of Dynamics to Physics and Chemistry, 47) that on dynamical principles there must be a reciprocal relation between the changes of dimensions produced by magnetization and the changes of magnetization attending mechanical strain.

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  • This was followed, in the second volume of the Miscellanea Taurinensia (1762) by his "Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies," together with the application of this important development of analysis to the solution of several dynamical problems, as well as to the demonstration of the mechanical principle of "least action."

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  • Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor theorems of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.

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  • The latter question can only be treated in connexion with the dynamical theory (see below, § but under all ordinary circumstances the result is independent of the precise answer that may be given.

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  • Although the matter can be fully treated only upon the basis of a dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens's principle to the calculation of the effects produced by opaque screens of limited extent.

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  • Dynamical Theory of Diffraction.

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  • This question was treated by Stokes in his " Dynamical Theory of Diffraction " (Camb.

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  • Cohnheim (1839-1884) and of Iliya Metchnikoff on the dynamical side of his- Fevers tology.

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  • Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.

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  • With the values above of u, v, w, u', v', w', the equations become of the form p x + 4 7rpAx -Fax -{-hy-}-gz =o, - p - dy+ 4?pBy + hx+ay+fz =o, P d p + TpCZ +f y + yz = o, and integrating p p 1+27rp(Ax2+By2+CZ2) +z ('ax e +ay e + yz2 2 f yz + 2gzx + 2 hx y) = const., (14) so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and 4c2 (c 2 - a2)?

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  • In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?

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  • Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

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  • When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express the fact that x, x, xs.

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  • the precedin investigation, the liquid stops dead when the body is brought to rest and when the body is in motion the surrounding liquid moves in uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on it surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations give above.

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  • 00 ab2dX °0 ab2dX (22) L (I JS)' (18) (t9) This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.

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  • Social physics will consist of the conditions and relations of the facts of society, and will have two departments, - one, statical, containing the laws of order; the other dynamical, containing the laws of progress.

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  • In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

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  • A number of molecules moving in obedience to dynamical laws will pass through a series of configurations which can be theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known.

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  • We begin with a general dynamical theorem, whose special application, when the dynamical system is identified with a gas, will appear later.

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  • qr,, be the generalized coordinates of any dynamical system, and let pi, P2,

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  • are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, v, w lie within ranges such that (i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds: (ii) E2mu2 E2mv 2 E2mw2 ?2aie12 s S s s s s - s E s' S' s' - - s' ' See Jeans, Dynamical Theory of Gases (1904), ch.

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  • It will at once be apparent that the kinetic theory of matter enables us to place the second law of thermodynamics upon a purely dynamical basis.

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  • The planetary theory and other particular dynamical problems likewise occupied his attention from time to time.

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  • Lamb, The Dynamical Theory of Sound (1910), is intended as a stepping-stone to the study of-the writings of Helmholtz and Rayleigh.

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  • These antinomies are four - two mathematical, two dynamical - connected with (I) the limitation of the universe in respect of space and time, (2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist), (3) the problem of freedom in relation to universal causality, (4) the existence of a universal being - about each of which pure reason contradicts the empirical, as thesis and antithesis.

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  • (2) An allowance is sometimes made for impact, that is the dynamical action of the live load due to want of vertical balance in the moving parts of locomotives, to irregularities of the permanent way, or to yielding of the structure.

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  • It has been repeatedly claimed for Mayer that he calculated the value of the dynamical equivalent of heat, indirectly, no doubt, but in a manner altogether free from error, and with a result according almost exactly with that obtained by J.

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  • The wider view, according to which the hypothesis of direct transmission of physical influences expresses only part of the facts, is that all space is filled with physical activity, and that while an influence is passing across from a body, A, to another body, B, there is some dynamical process in action in the intervening region, though it appears to the senses to be mere empty space.

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  • MacCullagh's hands the correct equations were derived from a single energy formula by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming.

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  • In this place it must suffice to indicate the gist of the more recent developments of the electro-optical theory, which involve the dynamical verification of Fresnel's hypothesis regarding optical convection and the other relations above described.

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  • The latter force is, by Maxwell's hypothesis or by the dynamical theory of an aether pervaded by electrons, the same as that which strair s the aether, and may be called the aethereal force; it thereby produces an aethereal electric displacement, say (f,g,h), according to the relation (f,g,h) = (41 r c 2) - 1(P',Q', RI), in which c is a constant belonging to the aether, which turns out to be the velocity of light.

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  • acting on the electrons situated on it which move with the velocity of the matter) is equal to minus the time-rate of change of the magnetic induction through that circuit as it moves with the matter, this being a dynamical consequence of the aethereal constitution assigned in (i).

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  • The dynamical series of stages in nature, the forms in which the ideal structure of nature is realized, are matter, as the equilibrium of the fundamental expansive and contractive forces; light, with its subordinate processes - magnetism, electricity, and chemical action; organism, with its component phases of reproduction, irritability and sensibility.'

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  • Just as nature exhibits to us the series of dynamical stages of processes by which spirit struggles towards consciousness of itself, so the world of intelligence and practice, the world of mind, exhibits the series of stages through which self-consciousness with its inevitable oppositions and reconciliations develops in its ideal form.

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  • Lord Rayleigh,' who has also investigated vibrating systems giving series of lines approaching a definite limit of " root," remarks that by dynamical reasoning we are always led to equations giving the square of the period and not the period, while in the equation representing spectral series the simplest results are obtained for the first power of the period.

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  • The discovery of the magnetic rotation of the plane of polarized light, though it did not lead to such important practical applications as some of Faraday's earlier discoveries, has been of the highest value to science, as furnishing complete dynamical evidence that wherever magnetic force exists there is matter, small portions of which are rotating about axes parallel to the direction of that force.

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  • A definition of the measurement dependent on dynamical theory has Time.

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  • Nevertheless, the theoretical possibility, and its realization in many cases, has brought considerations to the front which have recently become of predominant interest; consequently the possible transformations of isomers and polymers will be considered later under the denomination of reversible or dynamical isomerisms.

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  • Th~ axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions.

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  • But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another.

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  • (See also UNITS, DIMENSIONS or.) In any absolute system of dynamical measurement the fundamental units are those of mass, length and time we may denote them by the symbols M, L, T, respectively They may be chosen quite arbitrarily, e.g.

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  • In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios.

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  • The peculiar status of rigid bodies is that the principles in question are in most cases sufficient for the complete determination of the motion, the dynamical equations (I or 2) being equal in number to the degrees of freedom (six) of a rigid solid, whereas in cases where the freedom is greater we have to invoke the aid of other supplementary physical hypotheses (cf.

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  • For the determination of the motion it has only, been necessary to use one of the dynamical equations.

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  • qX +Pu = N.j These equations are applicable to any dynamical system whatever.

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  • When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are ~rn+qi=X, ~pl+rE=Y, ~f_qE+Pi,=Z, (II)

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  • As in the case of (2), the equations are applicable to any dynamical system whatever.

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  • Equations of Motion in Generalized Co-ordinates.Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces~ The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized coordinates of the system.

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  • These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated.

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  • Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical equations.

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  • Arago's previous investigations in the new science of electromagnetism, and crowned that labour by the announcement of his great discovery of the dynamical action between conductors conveying the electric currents.

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  • for 1865, entitled " A Dynamical Theory of the Electromagnetic Field."

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  • This fruitful conception, however, Bacon does not work out; and though he uses the word cause, and identifies form with formal cause, yet it is perfectly apparent that the modern notions of cause as dynamical, and of nature as in a process of flow or development, are foreign to him, and that in his view of the ultimate problem of science, cause meant causa immanens, or underlying substance, effects were not consequents but manifestations, and nature was regarded in a purely statical aspect.

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  • Invested, as society grows more complex, with a sanctity increasingly superior to that of the layman, the priest-king becomes the representative of the community as repository of its luck, whilst, as controller of all sacred forces that bear thereon, he is, as Dr Frazer puts it, " dynamical centre of the universe" (The Golden Bough (2nd ed.), i.

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  • When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x for the vapour, and po and Xo those for the liquid, x' - xo=JL - p(I/p' - I/pc),.

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  • (21) where J is the dynamical equivalent of heat, L is the latent heat of unit of mass of the vapour, and p is the pressure.

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  • The Detachment Of The Drop Is A Dynamical Effect, And It Is Influenced By Collateral Circumstances.

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  • But Although A Complete Solution Of The Dynamical Problem Is Impracticable, Much Interesting Information May Be Obtained From The Principle Of Dynamical Similarity.

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  • Dynamical Similarity Requires That T/Gva 2 Be Constant; Or, If G Be Supposed To Be So, That A' Varies As T/V.

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  • The coefficient (q) of the time in the exponential term (e at) may be considered to measure the degree of dynamical instability; its reciprocal 1 /q is the time in which the disturbance is multiplied in the ratio I: e.

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  • But they contain what is far more valuable still, the greatest addition which dynamical science had received since the grand strides made by Sir Isaac Newton and Joseph Louis Lagrange.

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  • Thus we feel it in its motion as wind, and observe the dynamical effects of this motion in the quiver of the leaf or the motion of a sailing ship. It offers resistance to the passage of bodies through it, destroying their motion and transforming their energy - as is betrayed to our hearing in the whiz of the rifle bullet, to our sight in the flash of the meteor.

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  • The practically obvious distinction between solids and fluids may be stated in dynamical language thus: - solids can sustain a longitudinal pressure without being supported by a lateral pressure; fluids cannot.

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  • The true foundations of a mechanical theory of the heavens were laid by Kepler's discoveries, and by Galileo's dynamical demonstrations; its construction was facilitated by the development of mathematical methods.

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  • The analytic method sought to express the moon's motion by integrating the differential equations of the dynamical theory.

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  • This was supplied by Fresnel, who, starting from a mechanical hypothesis, showed by ingenious but not strictly dynamical reasoning that if the incident stream have unit amplitude, that of the reflected stream will be - sin (i - r) /sin (i -{- r) or tan (i - r) /tan (i -{- r), according as the incident light is polarized in or perpendicularly to the plane of incidence i, r, being the angles of incidence and refraction connected by the formula sin i =,u sin r.

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  • The interdependence of motion and force was not indeed formulated into definite laws by Galileo, but his writings on dynamics are everywhere suggestive of those laws, and his solutions of dynamical problems involve their recognition.

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  • At first sight this period seemed to be inconsistent with dynamical theory.

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  • These changes arise from two causes - the one statical, the other dynamical.

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  • The dynamical causes are atmospheric and oceanic currents.

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  • Joule's views of the nature of heat strongly influenced Thomson's mind, with the result that in 1848 Thomson proposed his absolute scale of temperature, which is independent of the properties of any particular thermometric substance, and in 1851 he presented to the Royal Society of Edinburgh a paper on the dynamical theory of heat, which reconciled the work of N.

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  • Mayer and Joule, and placed the dynamical theory of heat and the fundamental principle of the conservation of energy in a position to command universal acceptance.

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  • According to his usual plan, Kant arranges these principles in conformity with the table of the categories, dividing the four classes, however, into two main groups, the mathematical and the dynamical.

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  • The mathematical principles are the abstract expression of the necessary mode in which data of sense are determined by the category in the form of intuitions or representations of objects; the dynamical are the abstract expression of the modes in which the existence of objects of intuition is determined.

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  • express determinations of the objects themselves; the dynamical are regulative, i.e.

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  • Under the dynamical principles, the general modes in which the existence of objects are determined, fall the analogies of experience, or general rules according to which the existence of objects in relation to one another can be determined, and the postulates of experience, the general rules according to which the existence of objects for us or our own subjective existence can be determined.

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  • This work involves studies of the Hybrid Monte Carlo algorithm for generating dynamical quarks.

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