where the functional symbols 4, 4, &c. denote any arbitrary functions what

ever.

59. It is then assumed that μ, v are the parameters of the confocal ellipses and hyperbolas situate in the moveable plane through the axis, viz. that we have

(the origin is midway between the two centres, 2b being their distance; , are in fact equal to the sum and difference u+v, u-v of the two centres respectively); and that the position of the moveable plane is determined by means of y, the inclination to a fixed plane through the axis, or say, as before, its azimuth. In fact, with these values of the coordinates, the expression of ds" is

which is of the required form. And moreover if the forces to the two centres vary as (dist.)-2, and there is besides a force to the middle point varying as the distance, then

whence (observing that λ=2-2) AU is of the required form. The equations obtained by substituting for U the above value give the ordinary solution of the problem.

60. Liouville's note to the last-mentioned memoir (1848) contains the demonstration of a theorem obtained by a different process in his second memoir, but which is in the present note, starting from Serret's formulæ, demonstrated by the more simple method of the first memoir, viz., it is shown that the motion can be obtained if the two centres, instead of being fixed, revolve about the point midway between them in a circle in such manner that the diameter through the two centres always passes through the projection of the body on the plane of the circle. It will be observed that the circular motion of the two centres is neither a uniform nor a given motion, but that they are, as it were, carried along with the moving body.

61. In Desboves's memoir "Sur le Mouvement d'un point matériel &c." (1848), the author developes the solution of the foregoing problem of moving centres, chiefly by the aid of the method employed in Liouville's second memoir. And he shows also that the methods of Euler and Lagrange for the case of two fixed centres apply with modification to the more complicated problem of the moving centres.

62. The problem of two centres is considered in Bertrand's "Mémoire sur les équations différentielles &c." (1852), by means of Jacobi's form of the

equations of motion, viz., the problem is reduced to a plane one by means of

the addition of a force α- (ante, No. 56).

1 y

63. Cayley's "Note on Lagrange's Solution &c." (1857) is merely a reproduction of the investigation in the Mécanique Analytique;' the object was partly to correct some slight errors of work, and partly to show what were the combinations of the differential equations, which give at once the integrals of the problem.

64. In § II. of Bertrand's "Mémoire sur quelques unes des formes &c." (1857), the following question is considered, viz., assuming that the dynamical equations

have an integral of the form

d2x dU day du
de dx' di dy'

=

=

a=Px'2+Qx'y'+Ry'2+Sy'+Tx'+K

(where a is the arbitrary constant, and P, Q... K are functions of a and y), it is required to find the form of the force-function U. It is found that U must satisfy a certain partial differential equation of the second order, the general solution of which is not known; but taking U to be a function of the distance from any fixed point (or rather the sum of any number of such functions), it is shown that the only case in which the differential equations for the motion of a point attracted to a fixed centre of forces have an integral of the form in question is the above-mentioned one of two centres, each attracting according to the inverse square of the distance, and a third centre midway between them, attracting as the distance.

The Spherical Pendulum, Article Nos. 65 to 73.

65. The problem is obviously the same as that of a heavy particle on the surface of a sphere.

I have not ascertained whether the problem was considered by Euler. Lagrange refers to a solution by Clairaut, Mém. de l'Acad. 1735.

The question was considered by Lagrange, Méc. Anal. 1st edit. p. 283. The angles which determine the position are the inclination of the string to the horizon, the inclination of the vertical plane through the string to a fixed vertical plane, or say the azimuth. And then forming the equations of motion, two integrals are at once obtained; these are the integrals of Vis Viva, and an integral of areas. And these give equations of the form dt=funct. (4) dy, dq=funct. (4)dy; so that t, o are each of them given by a quadrature in terms of 4, which is the point to which the solution is carried. It is noticed that may have a constant value, which is the case of the conical pendulum.

66. In the second edition, t. xi. p. 197 (1815), the solution is reproduced; only, what is obviously more convenient, the angles are taken to be

It is remarked that will always lie between a greatest value a and a least value ẞ, and the integrals are transformed by introducing therein instead of the angle σ, which is such that

coscos a sin❜a+cos ẞ cosa,

by which substitution they assume a more elegant form, involving only the radical

√1+k2 (cos ẞ—cos a) cos 2σ,

where k is a constant depending on cos a, cos ß; and the integration is effected approximately in the case where cos ẞ-cos a is small.

M. Bravais has noticed, however, that by reason of some errors in the working out, Lagrange has arrived at an incorrect value for the angle, which is the apsidal angle, or difference of the azimuths for the inclinations a and ß: see the 3rd edition (1855), Note VII., where M. Bravais resumes the calculation, and he arrives at the value =(1+aß), a and ß being small.

Lagrange considers also the case where the motion takes place in a resisting medium, the resistance varying as velocity squared.

67. A similar solution to Lagrange's, not carried quite so far, is given in Poisson's Mécanique,' t. i. pp. 385 et seq. (2nd ed. 1833).

A short paper by Puiseux, "Note sur le Mouvement d'un point matériel sur une sphère" (1842), shows merely that the angle is >

π

2

68. The ulterior development of the solution consists in the effectuation of the integrations by the elliptic and Jacobian functions. It is proper to remark that the dynamical problem the solution whereof by such functions was first fairly worked out, is the more difficult one of the rotation of a solid body, as solved by Jacobi (1839), in completion of Rueb's solution (1834), post, Nos. 186 and 197.

69. In relation to the present problem we have Gudermann's memoir " De pendulis sphæricis &c." (1849), who, however, does not arrive at the actual expressions of the coordinates in terms of the time; and the perusal of the memoir is rendered difficult by the author's peculiar notations for the elliptic functions*.

70. It would appear that a solution involving the Jacobian functions was obtained by Durège, in a memoir completed in 1849, but which is still unpublished; see § XX. of his Theorie der elliptischen Functionen' (1861), where the memoir is in part reproduced. It is referred to by Richelot in the Note presently mentioned.

71. We have next Tissot's Thèse de Mécanique,' 1852, where the expressions for the variables in terms of the time are completely obtained by means of the Jacobian functions H, O, and which appears to be the earliest published one containing a complete solution and discussion of the problem.

72. Richelot, in the Note "Bemerkungen zur Theorie des Raumpendels " (1853), gives also, but without demonstration, the final expressions for the coordinates in terms of the time.

Donkin's memoir "On a Class of Differential Equations &c." (1855) contains (No. 59) an application to the case of the spherical pendulum.

73. The first part of the memoir by Dumas, "Ueber die Bewegung des Raumpendels," &c. (1855), comprises a very elegant solution of the problem of the spherical pendulum based upon Jacobi's theorem of the Principal Function (1837), and which is completely developed by the elliptic and Jacobian functions. The latter part of the memoir relates to the effect of the rotation of the Earth; and we thus arrive at the next division of the general subject.

* The mere use of sn., cn., dn. as an abbreviation of the somewhat cumbrous sinam., cosam., Aam. of the 'Fundamenta Nova' is decidedly convenient.

Motion as affected by the Rotation of the Earth, and Relative Motion in general. Article Nos. 74 to 85.

74. Laplace (Méc. Céleste, Book X. c. 5) investigates the equations for the motion of a terrestrial body, taking account of the rotation of the Earth (and also of the resistance of the air), and he applies them to the determination of the deviations of falling bodies, &c. He does not, however, apply them to the case of the pendulum.

75. We have also the memoir of Gauss, " Fundamental-gleichungen, &c." (1804): the equations ultimately obtained are similar to those of Poisson. I have not had the opportunity of consulting this memoir.

76. Poisson, in the "Mémoire sur le mouvement des Projectiles &c." (1838), also obtains the general equations of motion, viz. (omitting terms involving. n2), these may be taken to be

(see p. 20), where the axes of x, y, z are fixed on the Earth and moveable with it: viz., z is in the direction of gravity; x, y in the directions perpendicular to gravity, viz., y in the plane of the meridian northwards, a westwards; g is the actual force of gravity as affected by the resolved part of the centrifugal force; B is the latitude. There are some niceties of definition which are carefully given by Poisson, but which need not be noticed here.

77. Poisson applies his formula incidentally to the motion of a pendulum, which he considers as vibrating in a plane; and after showing that the time of oscillation is not sensibly affected, he remarks that upon calculating the force perpendicular to the plane of oscillation, arising from the rotation of the Earth, it is found to be too small sensibly to displace the plane of oscillation or to have any appreciable influence on the motion-a conclusion which, as is well known, is erroneous. He considers also the motion of falling bodies, but the memoir relates principally to the theory of projectiles.

78. That the motion of the spherical pendulum is sensibly affected by the rotation of the Earth is the well-known discovery of Foucault; it appears by his paper, "Démonstration Physique &c.," Comptes Rendus, t. xxxii. 1851, that he was led to it by considering the case of a pendulum oscillating at the pole; the plane of oscillation, if actually fixed in space, will by the rotation of the Earth appear to rotate with the same velocity in the contrary direction; and he remarks that although the case of a different latitude is more complicated, yet the result of an apparent rotation of the plane of oscillation, diminishing to zero at the equator, may be obtained either from analytical or from mechanical and geometrical considerations. Some other Notes by Foucault on the subject are given, 'Comptes Rendus,' t. xxxv. (1853).

79. An analytical demonstration of the theorem was given by Binet, Comptes Rendus,' t. xxxii. (1851), and by Baehr (1853). Various short papers on the subject will be found in the Philosophical Magazine,' and elsewhere.

80. In regard to the above-mentioned problem of falling bodies, we have a Note by W. S., Camb. and Dub. M. Journ. t. iii. (1848), containing some errors

which are rectified in a subsequent paper," Remarks on the Deviation of Falling Bodies," &c. t. iv. (1849), by Dr. Hart and Professor W. Thomson. 81. The theory of relative motion is considered in a very general manner in M. Quet's memoir, "Des Mouvements relatifs en général &c." (1853). Suppose that x, y, z are the coordinates of a particle in relation to a set of moveable axes; let ', n', ' be the coordinates of the moveable origin in reference d2 E' d2n' d2¿′′ to a fixed set of axes, and treating the accelerations

dť2' dť2' dť2

as if they were coordinates, let these, when resolved along the moveable axes, give u', v', w': suppose, moreover, that p, q, r denote the angular velocities of the system of the moveable axes (or axes of x, y, z) round the axes of x, y, and z respectively; u', v', w', p, q, r are considered as given functions of the time, and then, if

dq -X +p (rx—pz)—9 (qz—ry )+w', dt

it is shown that the equations of motion are to be obtained from the equation

Em [(u-X)dx+(v−Y)èy+(w—Z)dz]=0,

where dx, y, z are the virtual velocities of the particle m in the directions of the moveable axes. This equation is in fact obtained as a transformation of

which belongs to a set of fixed axes of E, n, .

82. The equations for the motion of a free particle are of course u=X, v=Y, w=Z. In the case where the moveable axes are fixed on the Earth, and moveable with it (the diurnal motion being alone attended to), these lead to equations for the motion of a particle in reference to the Earth, similar to those obtained by Gauss and Poisson. The formulæ are applied to the case of the spherical pendulum, which is developed with some care; and Foucault's theorem of the rotation of the plane of oscillation very readily presents itself. The general formulæ are applied to the relative motion of a solid body, and in particular to the question of the gyroscope; the memoir contains other interesting results.

83. The principal memoirs on the motion of the spherical pendulum, as affected by the rotation of the Earth, are those of Hansen, "Theorie der Pendelbewegung &c." (1853), which contains an elaborate investigation of all the physical circumstances (resistance of the air, torsion of the string, &c.) which can affect the actual motion, and the before-mentioned memoir by Dumas, “ Ueber der Bewegung des Raumpendels &c." (1855). The investigation is conducted by means of the variation of the constants; the integrals for the undisturbed problem were, as already noticed, obtained by means of Jacobi's Principal Function, that is, in a form which leads at once to the expressions for the variation of the constants; and the investigation appears to be carried out in a most elaborate and complete manner.

84. In concluding this part of the subject I refer to Mr. Worms's work, The Rotation of the Earth' (1862), where the last-mentioned questions