if sin The form of the formula is, it will be observed, similar to that for motion in a straight line (and, 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 <kc." (1861). 20. The theorem was also obtained by Lagrange in the memoir "Rccherches <fcc." (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 mnniero particuliere d'exprimer le temps &c." (1778), reproduced in Note V. of the second volume of the last edition (Bertrand's) of the 'Mecanique 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 etseq.; Pagani, "Demonstration (Tun theoreme ifcc." (1834); and (in connexion with Hamilton's principal function) by Sir W, B. Hamilton, "On a General Method <fcc." (1834), p. 282; Jacobi, "Zur Thcorie &c." (1837), p. 122; Cayley, " Note on the Theory of Elliptic Motion" (1856). 22. Connected with the problem of central forc^, we have Sir W. It. Hamilton's 'Hodograph,' which in the paper (Proc. R. Irish Acad. 1847) is denned, and the fundamental properties stated; viz. if in an orbit round a centre of force there bo 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 (vj>=h), the hodograph is the reciprocal polar of the orbit with respect to a circle described about the centre of force, radius = ijh. 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 produot of the perpendicular on the tangent into the corresponding radius vector is =h. If force oc (dist.)-2, the hodograph, qua 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 hodographicaUy describing the intercepted arcs (that is, the times of describing the corresponding elliptic arcs) will be equal. 24. Droop, "On the Isochronism <fcc." (1856), shows geometrically that the last-mentioned property is equivalent to Lambert's theorem; and an .analytical demonstration is also given, Caylev, "A demonstration of Sir "W. B. Hamilton's Theorem <fcc." (1857). See also Sir W. B. Hamilton's 'Lectures on Quaternions ' (1853), p. 614. 25. The laws of central force which have been thus far referred to, are force li r 0.r, OC-j, OC -5; and it has been seen that the case of a force P+-^ depends upon that of a force P, so that the motions for the forces Ar+—. and _ +_ r3 r* r B are deducible from thoso for the forces Ar and — respectively. Some other A A B C D laws of force, e. g. 3+Br, _ + _+_-)—are considered by Legendre, "Thcorie de3 Fonctions Elliptiques" (1825), bciug suck as lead to results M expressible by elliptic integrals, and also the law —, for which the result in T 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 (riven in Stader's memoir "De Orbitis »fcc." (1852), where the investigation is conducted by means of the formula? which give t and 0 in terms of r (ante, 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. 8G 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 piano (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, presont 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 equations diffe'rentielles de la Mecanique " (1852), Donkin's memoir " On a Class of Differential Equations <fco." (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 ante, No. 15; they involve as an auxiliary quantity the cxccntric anomaly u. 28. Consider first the equation g=u—e sin «, which connects the moan anomaly g with the excentric anomaly tt. Any function of u, and in particular u itself, and the functions ^ nu may be expanded in terms of g by means of Lagrange's theorem (Lagrange, 'Mem. de Berlin,' 1768-1769, "Thcorie des Fonctions," c. 16, and "Traite de la Resolution des equations Nume'riques," Note 11). 29. Considering next the equation tan if=\ tan i u, V 1—e which gives the true anomaly in terms of the excentric anomaly, then, by replacing the circular functions by their exponential values (a process employed by Lagrange, * Mem. de Berlin,' 1776), / can be expressed in terms of tt; viz. the result is where X=- ——— I — - ). .Henco if w, sin w, sin 2», <fcc. are e expressed in terms of the mean anomaly, / will be obtained in tho form f=g-\-a, scrios of multiple sines of g, tho coefficients of the different terms being given in the first instance as functions of e and X; and to complete tho development A. and its powers have to bo developed in powers of e. The solution is carried thus far in the 'Mecaniquo Analytiquo' (1788), and in the < Mecaniquo Celeste ' (1799). 30. Wo havo next Bessel's investigations in tho Berlin Memoirs for 1816, 1818, and 1824, and which are carried on mainly by means of the integral * C1" 2*1 =1 cos(Au—£sinu)<2K, and various properties are there obtained and applications made of this important transcendant. 31. Relating to this integral we havo Jacobi's memoir, "Formulae trans formationis &c." (1836), Liouville, "Sur l'integrale J**008 * (M—* ^a u) (1841), and Hansen's "Emittelung der absolutcn Storungen" (1843); the researches of Poisson in the ' Connaissanco des Temps' for 1825 and 1836 are closely connected with those of Bessel. 32. A very elegant formula, giving the actual expression of tho coefficients considered as functions of e and A, is given by Greatheed in the paper " Inves • tigation of tho Genoral Term &c." (1838); viz. this is where, after developing in powers of X, tho negative powers of X must bo rejocted, and tho term independent of X divided by 2. This result is extended to other functions of/, Cayley "On certain Expansions «fec." (1842). 33. An expression for the coefficient of the general term as a function of e only is obtained, Lefort, "Expression Numerique &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 tho combination of the developments of / in terms of u, and it in terms of g; and an equivalent result is obtained directly in this manner, Creedy, "General and Practical Solution &c." (1855). 34. Wo have also on the subject of these developments the very valuable and interesting researches of Hansen, contained in his 'Fundamcnta Nova, &c.' (1838), in tho memoir "Ermittelung der absolutcn Storungen &c." (1845), and in particular in the memoir "Entwickelung des Products &c." (1853). 35. But tho expression for the coefficient of the general term ^ rg in any of theso expansions is so complicated that it was desirable to have for tho coefficients corresponding to the values r=0,1, 2,3, . . . the finally redueed expressions in which the coefficient of each power of £ is given as a numerical fraction. Such formulae for the development of ^-—l^m TM*jf, where j is a general symbol, the expansion being carried as far as «7, were given, Leverrier, « Annates do l'Observatoire de Paris,' t. i. (1855). 36. And starting from these I deduced the results given in my "Tables of the Developments, &,c." (1861); viz. these tables give (x=~—lY (** *'), ((=)'• -(;y.(^.-(5ns*j-»^-t all carried to eT. 37. Tlie true anomaly / has been repeatedly calculated to a much greater extent, in particular by Schubert (Ast. Theorique, St. Pet. 1822), as far as The expression for I as far as e'3 is given in the same work, and that for log - as far as e" was calculated by Oriani, sco 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 bo the distance from node, <j> the inclination, and x the reduced distance from node, then cosz=cos^cos;i', from which we may derive z=x+ series of multiple sines of x. And there are, moreover, the questions connected with the development of the reciprocal distance of two particles—say (a3-\-a'a—2aa'cos0)~»—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 / in multiple cosines or sines of g, the coefficient of a term ^ vj, where t is very great, is at most equal in absolute value to a quantity of the form J~^(j^' A and X 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 « is less than 0-6627432, which is the least modulus of e for which the equations 2=m—«sintt, lsecosu 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 1862. o the "Mcmoire eur la Mecanique Celesto," read at Turin in J83J, but it is reproduced in the memoir " Considerations nouvelles sur les suites &c.,*' Mem. d'Anal. et da Phys. Math. t. i. (1840); and see also the memoirs in i Jipuville's Journal' by Puiseux, and his Note i. to vol. ii. of the 3rd ed. of tho 'Mecanique 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 tho 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 t£, the solution of the problem leads ultimately to equations of the form dr _ ds \dr ads pdr ads 75-71' d<-71+71' ^=71+71' where R and 8 are rational and integral functions (of the third or fourth degree, in the case of forces varying as (dist.)_s) of r, s respectively (but they are not in general the same functions of r, s respectively); X and p are simple rational functions of r, and /x and a simple rational functions of s; so that tho equations give by quadratures, the first of them the curve described in the axial plane, tho 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 case, where R and S are each of them of the third or the fourth order, the quadratures depend on elliptic integrals t; but on account of the presence in tho formulse of the two distinct radicals VS, Vj-i> it would appear that tho solution is not susceptible of an ulterior development by means of elliptic and Jacobian functionsf similar to those obtained in the problems of Rotation and the Spherical Pendulum. 44. It has just been noticed that when R, 8 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 tho form ^^=^i, R and S being V R V S • If v, « are the distances of the body P from the centres A and B, a tho distance AT!, iy t> the angles at A and B respectively, andp*= tan \ { tan i Ij, 2=tani£-rtani q, then, as may be shown without difficulty, v+u = a j~~> e—w^a^^i 60 P m^ 2 are functions of v+« and v —« respectively; these quantities p and q are Eider's original coordinates. t The elliptic integrals aro Legendrc's functions F, E, n ; the elliptic and Jacobian functions are sinani., cosain.^ Aam., and the higher transoendanta 6, H. |