m! 
Mr. A. Cayley on the Equipotential Curve = + ah 145 
smaller one, +B’, does not enclose M’, but encloses the oval B 
which encloses M ; the curve consists of the exterior oval AA’, 
the ovals 7B! and mB! which have arisen out of the oval B’, and 
the oval B. As k& further decreases, the ovals AA! and 7B! con- 
tinually increase in magnitude, and the ovals mB! and B approxi- 
mate more and more nearly together; and at length, when & 
becomes =O, the ovals AA! and /B! disappear at infinity, while 
the ovals mB! and B unite themselves into a circle enclosing M, 
but not enclosing M’: the equation of this circle is, in fact, 
m mn 
2 
= + > ia ; or what is the same thing, 7?= a 7’?, and the points 
M, M’ have, in relation to this circle, the well-known relation 
that each is the image of the other. 
The preceding description is, I think, intelligible without the 
assistance of a series of figures illustrating the different forms of 
the curve, but there is no difficulty in actually tracing the curve 
for any particular values of the constant parameters. Thus 
(taking the distance MM! for unity) suppose that the equation 
of the curve is se a =1.2. (The value 1.2 was selected as 
a value not far from that for which the oval B! becomes a re- 
entrant figure of eight, though the change of form is so rapid that 
this value shows only the incipient tendency of the oval B! to 
take the form in question.) The form of the portion of the 
curve consisting of the two ovals B, B! will be that shown by 
the figure, which was constructed by points on a double scale 
with some accuracy. 
Phil, Mag. 8. 4. Vol. 14. No. 91, Aug. 1857. L 
