of Edinburgh, Session 1869 - 70 . 
93 
drawn through them, which is obviously the branch sought. But if 
A, B lie without the line, and 0, D on it, a circle through A, B hav¬ 
ing its centre 0 in CD produced, so that OA is a mean proportional 
between OC and OD is the circle required. If ABCD are all on a 
straight line, the other branch is manifestly a circle with centre on 
the line. 
Conics .—The parabola is an impossible conic for any finite num¬ 
ber of points. For the parabola has two asymptotes meeting at 
infinity. Hence the centre of gravity of an incomplete system, or 
of the sinks and sources separately in a complete system, must 
beat an infinite distance, which is absurd. The conics are there¬ 
fore central. 
The hyperbola , which has two asymptotes, is only possible when 
the cubic reduces to a conic. This demands that the centre of 
gravity of sinks and sources shall coincide, i.e., that AB, CD are 
diagonals of a parallelogram. The asymptotes must meet at right 
angles, and the hyperbola is equilateral. It is obvious, indeed, 
that in this case the sources and sinks give separately sets of con¬ 
centric rectangular hyperbolas, of which the one passing through 
the four points belongs to both sets, and is the only asymptotic 
curve of the complete system. 
In this case the equipotential lines are lemniscates. Let the 
origin be the centre of the system, 2 a and 2 b the diagonals of the 
parallelogram, a and /3 their angles with the initial line. At any 
point P 
AP 2 . BP 2 + ACP 2 . DP 2 = 0. 
That is. 
r 4 + a i — 2aV 2 cos 2 0 — a + A(r 4 + 6 4 - 26V 2 cos 2 6 - (3) — 0 
(1 -f- A)(r 4 + a 4 ) - 2r 2 cos 26 ( a 2 cos 2a + A6 2 cos 2(3) 
+ 2r 2 sin 26 (a 2 sin 2a + A6 2 sin 2(3) - 0 . 
cl 1 sin 2a 
When 
A = - 
the curve becomes 
6 2 sin 2(3 ’ 
(6 2 sin 2(3 - a 2 sin 2a)(r 4 + a 4 ) — 2a 2 6V 2 sin 2 (J3 — a) cos 26 = 0, 
a lemniscate, with foci on the initial line, and centre at the origin. 
If the parallelogram is a rectangle a ~ b, and the curve is 
i A — 2aV 2 CQS ^ a cos 26* -f a 4 = 0. 
cos fS 4 _ a 
VOL. VII. 
N 
