MATHEMATICAL SECTION. 
101 
ing about a centre under the action of central forces whose potential is 
a function of the sum of the squares of the radii vectores. 
The differential equations of this problem, in the case where the 
radii are supposed to describe no areas, were first integrated by 
Binet.* But the addition to the forces of the terms arising from cen¬ 
trifugal action much enhances the interest of the problem. 
The chief point of interest brought out by the solution was that 
while the directions of the points, whether as seen from the centre 
or from each other, all return to the same values after the lapse of 
a certain time, as do also the ratios of the radii vectores, the absolute 
values of the latter have all a factor whose period is generally dif¬ 
ferent from the former. Thus the movement of the system may be 
conceived as taking place under the operation of two distinct causes, 
viz: the first producing a revolution of all the points about the 
centre in closed curves in the same time, while the second, having a 
different period, changes the scale of representation of the system in 
space. 
[This paper appeared in full in the Annals of Mathematics. 4°. Char¬ 
lottesville, Va. 1887, October; vol. 3, no. 5, pp. 145-153.] 
Mr. Hill’s paper was briefly discussed by Mr. A. Hall and the 
Chairman. 
Mr. A. Hall read a paper on 
EULER’s THEOREM (GENERALLY CALLED LAMBERT’s). 
[Abstract.] 
This theorem is well known to astronomers and is very useful in 
computing the orbits of comets. The time of the passage of the 
comet from one point to another of its orbit is expressed by means 
of the two radii vectores drawn to the points, and the chord joining 
these points. For many years, and by many writers even of the 
present time, this theorem is attributed to Lambert. Mr. Hall 
stated that it was first given by Euler about 1743. He gives two 
proofs for the parabola, and then extends the theorem to the 
ellipse. For this case Euler gives first an expression for the peri¬ 
helion distance in terms of the sum of the two radii vectores and 
the chord. He then gives an approximate expression for the time 
of describing the arc in terms of the perihelion distance, the radii 
* Journal de mathematiques; par J. Liouville. 4°. Paris, 1837. 1st 
series, vol. 2, p. 457. 
