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The form of the formula is, it will be observed, similar to that for motion in a straight line (antè, No. 4), and in fact the motion in the ellipse is, by an ingenious geometrical transformation, made to depend upon that in the straight line. The geometrical theorems upon which the transformation depends are stated, Cayley "On Lambert's Theorem &c." (1861).

20. The theorem was also obtained by Lagrange in the memoir "Recherches &c." (1767) as a corollary to his solution of the problem of two centres; viz. upon making the attractive force of one of the centres equal to zero, and assuming that such centre is situate on the curve, the expression for the time presents itself in the form given by Lambert's theorem.

21. Two other demonstrations of the theorem are given by Lagrange in the memoir "Sur une manière particulière d'exprimer le temps &c." (1778), reproduced in Note V. of the second volume of the last edition (Bertrand's) of the Mécanique Analytique.' As M. Bertrand remarks, these demonstrations are very complete, very elegant, and very natural, assuming that the theorem is known beforehand.

Demonstrations were also given by Gauss, "Theoria Motus" (1809), p. 119 et seq.; Pagani, "Démonstration d'un théorème &c." (1834); and (in connexion with Hamilton's principal function) by Sir W. R. Hamilton, “On a General Method &c." (1834), p. 282; Jacobi, "Zur Theorie &c." (1837), p. 122; Cayley, "Note on the Theory of Elliptic Motion" (1856).

22. Connected with the problem of central forces, we have Sir W. R. Hamilton's Hodograph,' which in the paper (Proc. R. Irish Acad. 1847) is defined, and the fundamental properties stated; viz. if in an orbit round a centre of force there be taken on the perpendicular from the centre on the tangent at each point, a length equal to the velocity at that point of the orbit, the extremities of these lengths will trace out a curve which is the hodograph. As the product of the velocity into the perpendicular on the tangent is equal to twice the area swept out in a unit of time (vp=h), the hodograph is the reciprocal polar of the orbit with respect to a circle described about the centre of force, radius h. Whence also the tangent at any point of the hodograph is perpendicular to the radius vector through the corresponding point of the orbit, and the product of the perpendicular on the tangent into the corresponding radius vector is =h.

If force (dist.)-2, the hodograph, quà reciprocal polar of a conic section with respect to a circle described about the focus, is a circle.

23. The following theorem is also given without demonstration; viz. if two circular hodographs, which have a common chord passing or tending through a common centre of force, be both cut at right angles by a third circle, the times of hodographically describing the intercepted arcs (that is, the times of describing the corresponding elliptic arcs) will be equal.

24. Droop, "On the Isochronism &c." (1856), shows geometrically that the last-mentioned property is equivalent to Lambert's theorem; and an analytical demonstration is also given, Cayley, "A demonstration of Sir W. R. Hamilton's Theorem &c." (1857). See also Sir W. R. Hamilton's 'Lectures on Quaternions' (1853), p. 614.

25. The laws of central force which have been thus far referred to, are force 1 C α ; and it has been seen that the case of a force P+ depends

αν, α

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upon that of a force P, so that the motions for the forces Ar+and+

are deducible from those for the forces Ar and

A. B C. D

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laws of force, e. g. ±Br, A++ +-+ are considered by Legendre, "Théorie des Fonctions Elliptiques" (1825), being such as lead to results expressible by elliptic integrals, and also the law for which the result in

M

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volves a peculiar logarithmic integral. But the most elaborate examination of the different cases in which the solution can be worked out by elliptic integrals or otherwise is given in Stader's memoir "De Orbitis &c." (1852), where the investigation is conducted by means of the formula which give t and in terms of r (antè, No. 14).

26. In speaking of a central force, it is for the most part implied that the force is a function of the distance: for some problems in which this is not the case, see post, Miscellaneous Problems, Nos. 86 and 87.

It is to be noticed that, although the problem of central forces may be (as it has so far been) considered as a problem in plano (viz. the plane of the motion has been made the plane of reference), yet that it is also interesting to consider it as a problem in space; in fact, in this case the integrals, though of course involved in those which belong to the plane problem, present themselves under very distinct forms, and afford interesting applications of the theory of canonical integrals, the derivation of the successive integrals by Poisson's method, and of other general dynamical theories. Moreover, in the lunar and planetary theories, the problem must of necessity be so treated. Without going into any details on this point, I will refer to Bertrand's memoir "Sur les équations différentielles de la Mécanique" (1852), Donkin's memoir "On a Class of Differential Equations &c." (1855), and Jacobi's pos- . thumous memoir, "Nova Methodus &c." (1862).

Elliptic Motion, Article Nos. 27–40.

27. The question of the development of the true anomaly in terms of the mean anomaly (Kepler's problem), and of the other developments which present themselves in the theory of elliptic motion, is one that has very much occupied the attention of geometers. The formula on which it depends are mentioned antè, No. 15; they involve as an auxiliary quantity the excentric anomaly u.

28. Consider first the equation

g=u-e sin u,

which connects the mean anomaly g with the excentric anomaly u.

COS

sin
nu may

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Any function of u, and in particular u itself, and the functions be expanded in terms of g by means of Lagrange's theorem (Lagrange, Mém. de Berlin,' 1768-1769, "Théorie des Fonctions," c. 16, and "Traité de la Résolution des équations Numériques," Note 11). 29. Considering next the equation

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which gives the true anomaly in terms of the excentric anomaly, then, by replacing the circular functions by their exponential values (a process em

ployed by Lagrange, 'Mém. de Berlin,' 1776), ƒ can be expressed in terms of u; viz. the result is

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expressed in terms of the mean anomaly, ƒ will be obtained in the form. f=g+a series of multiple sines of g, the coefficients of the different terms being given in the first instance as functions of e and A; and to complete the development A and its powers have to be developed in powers of e. The solution is carried thus far in the Mécanique Analytique' (1788), and in the 'Mécanique Céleste' (1799).

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30. We have next Bessel's investigations in the Berlin Memoirs for 1816, 1818, and 1824, and which are carried on mainly by means of the integral

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and various properties are there obtained and applications made of this important transcendant.

31. Relating to this integral we have Jacobi's memoir, "Formulæ trans

formationis &c.” (1836), Liouville, "Sur l'intégrale ("

S

cos i (u-x sin u) du” (1841), and Hansen's "Ermittelung der absoluten Störungen" (1843); the researches of Poisson in the Connaissance des Temps' for 1825 and 1836 are closely connected with those of Bessel.

32. A very elegant formula, giving the actual expression of the coefficients considered as functions of e and A, is given by Greatheed in the paper " Investigation of the General Term &c." (1838); viz. this is

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where, after developing in powers of A, the negative powers of λ must be rejected, and the term independent of A divided by 2. This result is extended to other functions of f, Cayley "On certain Expansions &c." (1842). 33. An expression for the coefficient of the general term as a function of e only is obtained, Lefort, "Expression Numérique &c." (1846). The expression, which, from the nature of the case, is a very complicated one, is obtained by means of Bessel's integral. This is an indirect process which really comes to the combination of the developments of ƒ in terms of u, and u in terms of g; and an equivalent result is obtained directly in this manner, Creedy, "General and Practical Solution &c." (1855).

34. We have also on the subject of these developments the very valuable and interesting researches of Hansen, contained in his Fundamenta Nova, &c.' (1838), in the memoir "Ermittelung der absoluten Störungen &c." (1845), and in particular in the memoir "Entwickelung des Products &c." (1853).

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sin

35. But the expression for the coefficient of the general term rg in any of these expansions is so complicated that it was desirable to have for the coefficients corresponding to the values r=0, 1, 2, 3, . . . the finally reduced expressions in which the coefficient of each power of e is given as a numerical

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fraction. Such formulæ for the development of (-1)

m

(2-1)" con jf, where

sin

j is

a general symbol, the expansion being carried as far as e, were given, Leverrier, Annales de l'Observatoire de Paris,' t. i. (1855).

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36. And starting from these I deduced the results given in my "Tables of

the Developments, &c.” (1861); viz. these tables give (x=2—1),

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-1

((); ·()', () ̄`· ·(1) ̄) conf, j=1 to j=7,

a

all carried to e".

a

37. The true anomaly ƒ has been repeatedly calculated to a much greater extent, in particular by Schubert (Ast. Théorique, St. Pét. 1822), as far as e2o. The expression for as far as e13 is given in the same work, and that

for log

a

g
a

as far as e was calculated by Oriani, see Introd. to Delambre's 'Tables du Soleil,' Paris (1806).

38. It may be remarked that when the motion of a body is referred to a plane which is not the plane of the elliptic orbit, then we have questions of development similar in some measure to those which regard the motion in the orbit; if, for instance, z be the distance from node, & the inclination, and the reduced distance from node, then cos z=cos o cosa, from which we may derive z=x+ series of multiple sines of a. And there are, moreover, the questions connected with the development of the reciprocal distance of two particles-say (a+a2-2aa' cos 0)--which present themselves in the planetary theory; but this last is a wide subject, which I do not here enter upon. I will, however, just refer to Hansen's memoir, " Ueber die Entwickelung der negativen und ungeraden Potenzen &c." (1854).

39. The question of the convergence of the series is treated in Laplace's memoir of 1823, where he shows that in the series which express r and ƒ in

COS

multiple cosines or sines of g, the coefficient of a term ig, where i is very

sin

great, is at most equal in absolute value to a quantity of the form

A

(

Wi A and A being finite quantities independent of i, whence he concludes that, in order to the convergency of the series, the limiting value of the excentricity is e=A, the numerical value being e=0·66195.

40. The following important theorem was established by Cauchy, as part of a theory of the convergence of series in general; viz. so long as e is less than 0-6627432, which is the least modulus of e for which the equations

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can be satisfied, the development of the true anomaly and other developments in the theory of elliptic motion will be convergent. This was first given in

the "Mémoire sur la Mécanique Céleste," read at Turin in 1831, but it is reproduced in the memoir " Considérations nouvelles sur les suites &c.," Mém. d'Anal. et de Phys. Math. t. i. (1840); and see also the memoirs in Liouville's Journal' by Puiseux, and his Note i. to vol. ii. of the 3rd ed. of the Mécanique Analytique' (1855). There are on this subject, and on subjects connected with it, several papers by Cauchy in the Comptes Rendus,' 1840 et seq., which need not be particularly referred to.

The Problem of two Centres, Article Nos. 41 to 64.

41. The original problem is that of the motion of a body acted upon by forces tending to two centres, and varying inversely as the squares of the distances; but, as will be noticed, the solutions apply with but little variation to more general laws of force.

42. It may be convenient to notice that the coordinates made use of (in the several solutions) for determining the position of the body, are either the sum and difference of the two radius vectors, or else quantities which are respectively functions of the sum and the difference of these radius vectors*. If the plane of the motion is not given, then there is a third coordinate, which is the inclination of the plane through the body and the two centres to a fixed plane through the two centres, or say the azimuth of the axial plane, or simply the azimuth.

43. Calling the first-mentioned two coordinates r and s, and the azimuth, the solution of the problem leads ultimately to equations of the form

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pdr, ods
d = √ R
+
WR S

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where R and S are rational and integral functions (of the third or fourth degree, in the case of forces varying as (dist.)-2) of r, s respectively (but they are not in general the same functions of r, s respectively); λ and ρ are simple rational functions of r, and μ and o simple rational functions of s; so that the equations give by quadratures, the first of them the curve described in the axial plane, the second the position of the body in this curve at a given time, and the third of them the position of the axial plane. In the ordinary ease, where R and S are each of them of the third or the fourth order, the quadratures depend on elliptic integrals+; but on account of the presence in the formula of the two distinct radicals R, S, it would appear that the solution is not susceptible of an ulterior development by means of elliptic and Jacobian functions+ similar to those obtained in the problems of Rotation and the Spherical Pendulum.

44. It has just been noticed that when R, S are each of them of the fourth order, the quadratures depend on elliptic integrals; in the particular cases

in which the relation between r, s is of the form

mdr nds
VR VS

=

R and S being

*If v, u are the distances of the body P from the centres A and B, a the distance AB, ,n the angles at A and B respectively, and p=tan tan n, q=tan÷tan, then,

1+p 1-p'

as may be shown without difficulty, v+u=a v-u=a

1-q
1+q'

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functions of v+u and v-u: respectively; these quantities p and q are Euler's original coordi

nates.

The elliptic integrals are Legendre's functions F, E, II; the elliptic and Jacobian functions are sinam., cosam., Aam., and the higher transcendants O, H.

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