ON THE SPECIAL PROBLEMS OF DYNAMICS. 199 
motion is oscillatory, and which, as he remarks, seem to furnish the first 
instance of the description by a free particle of only a finite portion of the 
curve which is analytically the orbit of the particle ; there is, however, nothing 
surprising in this kind of motion, although its existence might easily not have 
been anticipated. 
56. § 26 of Jacobi’s memoir “ Theoria Novi Multiplicatoris &c.” (1845) is 
entitled ‘Motus puncti versus duo centra secundum legem Neutonianum 
attracti.” The equations for the motion in space are by a general theorem 
given in the memoir “ De Motu puncti singularis ” (1842), reduced to the case 
of motion in a plane: viz. if w, y are the coordinates, the centre point of the 
axis being the origin, and y being at right angles to the axis, andif the distance 
ay 
dt? 
2 
there is added a term ra which arises from the rotation about the axis. Two 
of the centres is 2a; then the only difference is that to the expression for 
integrals are obtained, one the integral of Vis Viva, and the other of them an 
integral similar to one of those of Euler’s or Lagrange’s. And then 2’, y’ 
being the differential coefficients of w, y with regard to the time, the remain- 
ing equation may be taken to be y'dv—a‘dy=0, where wx’, y' are to be 
expressed as functions of w, y by means of the two given integrals. This 
being so, the principle of the Ultimate Multiplier * furnishes a multiplier of 
this differential equation, and the integral is found to be 
y'du—x'dy 
ay (#@—y")+ (Ca +y)ay © 
the quantity under the integral sign being a complete differential. To verify 
a@ posteriori that this is so, Jacobi introduces the auxiliary quantities X’, \" 
defined as the roots of the equation \°+A(a*+y’?—a’*)—a’y’?=0, which in 
fact, if as before u, v are the distances from the centres, leads to 
u+-v=2V PN, u—v=2V aX", 
so that \’,” are functions of w+v, w—v respectively ; and the formulz, 
as ultimately expressed in terms of X’, X”, are substantially of the same form 
with those of Euler and Lagrange. 
57. The investigations contained in Liouville’s three memoirs “ Sur quel- 
ques cas particuliers &c.” (1846), find their chief application in the problem 
of two centres, and by leading in the most direct and natural manner to the 
general law of force for which the integration is possible, they not only give 
some important extension of the problem, but they in fact exhibit the pro- 
blem itself and the preceding solutions of it in their true light. But as they 
do not relate to this problem exclusively, it will be convenient to consider 
them separately under the head Miscellaneous Problems. 
58. In Serret’s ‘ Thése sur le Mouvement &c.’ (1848), the problem is very 
elegantly worked out according to the principles of Liouville’s memoirs as 
follows: viz. assuming that the expression of the distance between two con- 
secutive positions of the body is 
i ds? =(mdp? +ndy*)+Xr"dy’, 
where m, n are functions of , v respectively, and if the forces can be repre- 
sented by means of a force-function U, then the motion can be determined, 
* Explained in Jacobi’s memoir “Theoria Novi Multiplicatoris &e.,” Crelle, tt. xxvii. 
XXvill. xxix. 1844-465. : : 
