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النشر الإلكتروني

Rotation of a Solid Body;

Kinematics of a Solid Body;

Miscellaneous Problems.

As regards the first division of the subject, I remark that the lunar and planetary theories, as usually treated, do not (properly speaking) relate to the problem of three bodies, but to that of disturbed elliptic motion—a problem which is not considered in the present Report. The problem of the spherical pendulum is that of a particle moving on a spherical surface; but, with this exception, I do not much consider the motion of a particle on a given curve or surface, nor the motion in a resisting medium; what is said on these subjects is included under the head Miscellaneous Problems. The first six heads relate exclusively, and the head Miscellaneous Problems relates principally to the motion of a single particle. As regards the second division of the subject, I will only remark that, from its intimate connexion with the theory of the motion of a solid body, I have been induced to make a separate head of the geometrical subject, "Transformation of Coordinates," and to treat of it in considerable detail.

I have inserted at the end of the present Report a list of the memoirs and works referred to, arranged (not, as in the former Report, in chronological order, but) alphabetically according to the authors' names: those referred to in the former Report formed for the purpose thereof a single series, which is not here the case. The dates specified are for the most part those on the titlepage of the volume, being intended to show approximately the date of the researches to which they refer, but in some instances a more particular specification is made.

I take the opportunity of noticing a serious omission in my former Report, viz., I have not made mention of the elaborate memoir, Ostrogradsky, "Mémoire sur les équations différentielles rélatives au problème des Isopérimètres," Mém. de St. Pét. t. iv. (6 sér.) pp. 385-517, 1850, which among other researches contains, and that in the most general form, the transformation of the equations of motion from the Lagrangian to the Hamiltonian form, and indeed the transformation of the general isoperimetric system (that is, the system arising from any problem in the calculus of variations) to the Hamiltonian form. I remark also, as regards the memoir of Cauchy referred to in the note p. 12 as an unpublished memoir of 1831, there is an "Extrait du Mémoire présenté à l'Académie de Turin le 11 Oct. 1831," published in lithograph under the date Turin, 1832, with an addition dated 6 Mar. 1833. The Extract begins thus :-" § I. Variation des Constantes Arbitraires. Soient données entre la variable t, . n fonctions de t désignées par x, y, z . . et n autres fonctions de t désignées par u, v, w, . . 2n équations différentielles du prémier ordre et de la forme

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dQ dv dQ dw dQ

==

dx' dt dy' dt

&c." "dz'

without explanation as to the origin of these equations; and the formulæ are then given for the variations of the constants in the integrals of the foregoing system; this seems sufficient to establish that Cauchy in the year 1831 was familiar with the Hamiltonian form of the equations of motion.

Bour's "Mémoire sur l'intégration des équations différentielles de la Mécanique," as published, Mém. prés. de l'Inst. t. xiv. pp. 792-821, is substan

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tially the same as the extract thereof in Liouville's Journal,' referred to in my former Report; but since the date of that Report there have been published in the Comptes Rendus,' 1861 and 1862, several short papers by the same author; also Jacobi's great memoir, see list, Jacobi, Nova Methodus &c. 1862, edited after his decease by Clebsch; some valuable memoirs by Natani and Clebsch (Crelle, 1861 and 1862) relating to the Pfaffian system of equations. (which includes those of Dynamics), and Boole " On Simultaneous Differential Equations of the First Order, in which the number of the Variables exceeds by more than one the number of the Equations," Phil. Trans. t. clii, (1862) pp. 437-454.

Rectilinear Motion, Article Nos. 1 to 5.

1. The determination of the motion of a falling body, which is the case of a constant force, is due to Galileo.

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2. A variable force, assumed to be a force depending only on the position of the particle, may be considered as a function of the distance from any point in the line, selected at pleasure as a centre of force; but if, as usual, the force is given as a function of the distance from a certain point, it is natural to take that point for the centre of force. The problem thus becomes a particular case of that of central forces; and it is so treated in the Principia,' Book I. §7; the method has the advantage of explaining the paradoxical result which presents itself in the case Force ∞ (Dist.)-2, and in some other eases where the force becomes infinite. According to theory, the velocity becomes infinite at the centre, but the direction of the motion is there abruptly reversed; so that the body in its motion does not pass through the centre, but on arriving there, forthwith returns towards its original position; of course such a motion cannot occur in nature, where neither a force nor a velocity ever is actually infinite.

3. Analytically the problem may be treated separately by means of the

d2x

equation=X, which is at once integrable in the form ()=C+2/Xdx.

4. The following cases may be mentioned :—

Force Dist. The law of motion is well known, being in fact the same as for the cycloidal pendulum.

Force (Dist.)-2, μ
=

which is the case above alluded to.

Assuming that the body falls from rest at a distance a, we have

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x=a (1—cos 4),

, is given in terms of the time by means of the equation

nt=p-sin p.

If the body had initially a small transverse velocity, the motion would be in a very excentric ellipse, and the formulæ are in fact the limiting form of those for elliptic motion.

5. There are various laws of force for which the motion may be determined. In particular it can be determined by means of Elliptic Integrals, in the case of a body attracted to two centres, force ∞ (dist.)-2: see Legendre, Exercices de Cal. Intég. t. ii. pp. 502-512, and Théorie des Fonct. Ellip. t. i. pp. 531– 538.

Central Forces, Article Nos. 6 to 26.

6. The theory of the motion of a body under the action of a given central force was first established in the Principia,' Book I. §§ 2 & 3: viz. Prop. I. the areas are proportional to the times, that is (using the ordinary analytical

1.

notation), r2d0=hdt, and Prop. VI. Cor. 3, PSY. PV

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7. It is to be noticed that, given the orbit, the law of force is at once determined; and § 2 contains several instances of such determination; thus, Prop. VII. If a body revolve in a circle, the law of force to a point S is

1

force Sp2. Pys (P the body, PV the chord through $).

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PV3

Prop. IX. If a body move in a logarithmic spiral, force α (dist.)-3.

Prop. X. If a body move in an ellipse, force to centre a dist., and as a particular case, if the body move in a parabola under the action of a force parallel to the axis, the force is constant. The particular case of motion in a parabola had been obtained by Galileo.

And § 3. Props. XI. XII. XIII. If a body move in an ellipse, hyperbola, or parabola under the action of a force tending to the focus, force a (dist.)-2.

8. But Newton had no direct method of solving the inverse problem (which depends on the solution of the differential equation), "Given the force to find the orbit." Thus force α (dist.)-2, after it has been shown that an ellipse, a hyperbola, and a parabola may each of them be described under the action of such a force. The remainder of the solution consists in showing that, given the initial circumstances of the motion, a conic section (ellipse, parabola, or hyperbola, as the case may be) can be constructed, passing through the point of projection, having its tangent in the direction of the initial motion, and such that the velocity of the body describing the conic section under the action of the given central force is equal to the velocity of projection; which being so, the orbit will be the conic section so constructed. This is what is done, Prop. XVII.; it may be observed that the latus rectum is constructed not very elegantly by means of the latus rectum of an auxiliary orbit.

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9. A more elegant construction was obtained by Cotes (see the Harmonia Mensurarum,' pp. 103-105, and demonstration from the author's papers in the Notes by R. Smith, pp. 124, 125), depending on the position of a point C, such that the velocity acquired in falling under the action of the central force from C directly or through infinity* to P the point of projection, is equal to the given velocity of projection.

10. But Newton's original construction is now usually replaced by a construction which employs the space due to the velocity of projection considered as produced by a constant force equal to the central force at the point of projection.

11. Section 9 of Book I. relates to revolving orbits, viz., it is shown that a body may be made to move in an orbit revolving round the centre of force,

* In the second case C lies on the radius vector produced beyond the centre, and the body is supposed to fall from rest at C (under the action of the central force considered as repulsive) to infinity, and then from the opposite infinity (with an initial velocity equal to the velocity so acquired) to P.

by adding to the central force required to make the body move in the same orbit at rest, a force α (dist.)-3. This appears very readily by means of the differential equation (antè, No. 6), viz. writing therein P+cu3 for P, and then 0', 7' in the place of e√√1—1— respectively, the equation retains

h' 1.

с

h2

its original form, with 0', h', in the place of 0, h respectively.

12. It may be remarked that when the original central force vanishes, the fixed orbit is a right line (not passing through the centre of force). It thus appears by § 9 that the curve u=A cos (n0+B) may be described under the action of a force α (dist.)-3. A proposition in § 2, already referred to, shows that a logarithmic spiral may be described under the action of such a force.

13. But the case of a force α (dist.)-3 was first completely discussed by Cotes in the Harmonia Mensurarum,' pp. 31-35, 98-104, and Notes, pp. 117 -173. There are in all five cases, according as the

velocity of projection is

1. Less than that acquired in falling from infinity, or say equal to that acquired in falling from a point C to P, the point of projection.

2. Equal to that acquired in falling from infinity.

3, 4, 5. Greater than that acquired in falling from infinity, or say equal to that acquired in falling from a point C', through infinity, to P; viz. PQ being the direction of projection, and SQ, C'T perpendiculars thereon from S and C' respectively,

3. SQ<TQ;

4. SQ=TQ;

S

P

5. SQ>TQ;

T

the equations of the orbits being

1. u=Aeme

'+Be-me, A and B same sign, so that rad. vector is never infinite.

2. u=Aeme or Be-me, logarithmic spiral.

3. u=Aemo+Be-me, A and B opposite signs, so that rad. ector becomes infinite.

4. u=A0+B, m=0, reciprocal spiral.

5. u=A cos (no+B), m=n√=1.

14. The before-mentioned equation,

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is in effect given (but the equation is encumbered with a tangential force) in Clairaut's "Théorie de la Lune," 1765. It is given in its actual form, and extensively used (in particular for obtaining the above-mentioned equations for Cotes's spirals) in Whewell's Dynamics,' 1823. The equation appears to be but little known to continental writers, and (under the form u"+u—a2r3R=0) it is given as new by Schellbach as late as 1853. The formulæ used in place of it are those which give t and each of them in terms of r; viz.

dt=

de:

=

rdr

{—h2+r2 (C—2f Pdr)}*

hdr

r{—h2+r2(C—2ƒ Pdr)}**

which, however, assume that P is a function of r only.

15. Force (dist.)-2. The law of motion in the conic sections is implicitly given by Newton's theorem for the equable description of the areas. For the parabola, if a denote the pericentric distance, and f the angle from pericentro or true anomaly, we have

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For the ellipse we have an angle g, the mean anomaly varying directly as

the time (g=nt if n=

απ

an auxiliary angle u, the excentric anomaly,

connected with g by the equation

g-u-e sin u;

and then the radius vector r and the true anomaly ƒ are given in terms of u by the equations r≈a (1—e cos u), and

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a

16. It is very convenient to have a notation for and ƒ considered as functions of e, g, and I have elsewhere proposed to write

ra elqr (e, g), f=elta (e, g),

read elqr elliptic quotient radius, and elta elliptic true anomaly.

17. The formula for the hyperbola correspond to those for the ellipse, but they contain exponential in the place of circular functions (see post, Elliptic Motion).

18. Euler, in the memoir "Determinatio Orbitæ Cometa Anni 1742,” (1743), p. 16 et seq., obtained an expression for the time of describing a parabolic are in terms of the radius vectors and the chord; viz. these being f, g, and k, the expression is

Time =

1

= 0 √ μ{ (S+9+2) * — (S+0−x)"},

which, however, as remarked by Lagrange, Méc. Anal.' t. xi. (3rd edit. p. 28), is deducible from Lemma X. of the third book of the Principia.' But the theorem in its actual form is due to Euler.

19. Lambert, in the Proprietates Insigniores, &c.' (1761), Theorem VII. Cor. 2, obtained the same theorem, and in section 4 he obtained the corresponding theorem for elliptic motion; viz. the expression for the time is

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