200 REPORT—1862, 
provided only , AU, = are of the forms 
A=gu—Oy, 
AU=yp—Wy, 
A =op— Ty, 
where the functional symbols ¢, @, &c. denote any arbitrary functions what- 
ever. 
59. It is then assumed that p, vy are the parameters of the confocal ellipses 
and hyperbolas situate in the moveable plane through the axis, viz. that we 
have 
“Stra: ammo 
2 oe pe b? ? 
2 2 
wv 
y b—pr 
=i 
(the origin is midway between the two centres, 2b being their distance ; 
3u, 4v 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 deter- 
mined 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 
2 2 l 2 if 5 nae lin es 2 
which is of the required form. And moreover if the forces to the two centres 
yary as (dist.)—?, and there is besides a force to the middle point varying as 
the distance, then 
U= ee at Fe —b*), 
p+y pov 
whence (observing that A=,?—»*) AU is of the required form, The equa- 
tions 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 formule, 
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 &e,” 
(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 
