7. Explain wherein consists the peculiarity of the following problem, and solve it
by geometrical considerations:—
The peculiarity of the problem is that the variable parameters upon which the
circle depends, (say a, /3 the coordinates of the centre and k the radius), are not
subject to any equations, but only to the inequalities
№>{a- aff + (/3 - A)' 2 ,
k 2 >(a- a,)' 2 + (/3 - /3 2 ) 2 ,
k 2 > (a - ot 3 ) 2 + (/3 - A) 2 ,
(a 1} A; a 2, Al A> the coordinates of the given points, and the sign > including =).
The problem therefore cannot be solved by the ordinary analytical method, but it is
easily solved geometrically as follows: Let A, B, C be the three points; consider all
the circles inclosing the three points, viz. 0 a circle not passing through any of them ;
A a circle through the point A, B a circle through the point B, AB a circle through
the points A and B, &c. Then for any circle 0, if the centre be fixed and the
radius gradually diminish, the circle will at last pass through one of the points ABC;
that is, every circle 0 is greater than some circle A, B, or C; and the circle 0 is
therefore not a minimum. Taking next a circle A, we may imagine the centre to move
from its original position in a straight line towards the point A, the circle thus
gradually diminishing until it passes through one of the points B or C; that is, every
circle A is greater than some circle AB or AG, and therefore no circle A is a minimum;
and in like manner no circle B or G is a minimum. There remain the circles
AB, AC, BG; if the triangle ABC is acute-angled, then in each series, the least circle
is the circle ABC circumscribed about the triangle; and this is then the minimum
circle inclosing the three points. But if the triangle is obtuse-angled, say at C, then
the least circle CA or GB is the circle ABC circumscribed about the triangle; but
this is not the least circle AB, viz. the circle AB, being diminished to ABC, may
be further diminished until it becomes the circle on the diameter AB; but below
this it cannot be diminished; and consequently the minimum circle inclosing the three
points is in this case the circle on the diameter AB.
8. A particle describes an ellipse under the simultaneous action of given central
forces, each varying as (distance) -2 , at the two foci respectively: find the differential
relation between the time and the eccentric anomaly.
Taking the equation of the ellipse to be ~ +1- 2 = 1, and the absolute forces at
the two foci (ae, 0), {—ae, 0) to be y, y respectively, the differential equations of
motion will be
d 2 x x — ae
dt 2 ^ 1
, (x + ae)
