ON THE SPECIAL PROBLEMS OF DYNAMICS. 211 
employed by the author, these are the semiaxes of the confocal ellipse and 
hyperbola represented by the equations 
ae y? 
Pa y 
we we Gb? ? pal prays ? 
—very interesting results are obtained. The equations give 
Bx? =r, By? = (y2—2*) (P—»?), 
and thence 
dy? dy’? 
which is of the proper form, and the corresponding expression of U is 
eh, Fu—Fv : 
ee Ens 5a 
so that the foree-function having this value (fu, Fy being arbitrary functions 
of » and » respectively), the equations of motion may be integrated by qua- 
dratures. 
103. In particular, if 
Sfu=getg'p+k(e—o'p’), 
Fry=qgv—q'y +k(v*—6b*r’), 
then 
Ven Feb +I 4 byt). 
pty poy 
But »+yv, w—y are the distances of the point from the two foci, and 
pe+r?—b?(=2"+7’) is the square of the distance from the centre, so that 
the expression for U is 
Bie ‘site ao 
el 
and the case is that of forces to the foci varying inversely as the squares of 
the distances, and a force to the centre varying directly as the distance— 
the case considered by Lagrange in the problem of two centres. But this is 
merely one particular case of those given by the general formula. 
The cases g=0, g'=0, k=0 (no forces), and g=0, g!=0 (a force to the 
centre) lead to some interesting results; it is noticed also that the expression 
funct. Signa aia! cama 8 ote) 
Tr 
that it may be thereby ascertained (without transforming to elliptic coordi- 
nates) whether a given value of the force-function is of the form considered 
in the theory. 
In § 3 the author considers the expression dz? +dy?=X(da? +d’), d being 
in the first instance any function whatever of @ and (§; and he shows that the 
expressions of xv, y are given by the equation 
x+y —1=V(ae+BV —1), 
w being any real function. If, however, it is besides assumed that d is of 
the required form=fa—F{, then he shows that the system of elliptic coordi- 
nates is the only one for which the conditions are satisfied. §§ 4,5, 6, and 7 
relate to the motion of a point on a sphere, an ellipsoid, a surface of revolu- 
tion, and the skew helicoid respectively ; and the concluding § 8 contains only 
a brief reference to the author’s second memoir. 
104. Liouyille’s second and third memoirs may be more briefly noticed. 
In the second memoir the author starts from Jacobi’s theorem of the V 
f P2 
for the force-function may be written U= 
