! 
Mr. A. Cayley on the Equipotential Curve - + or =e, Lae 
sider r, 7! as the distances in plano of a point P of the curve from 
the given points M, M’: such curve may be termed the equipo- 
tential curve. I propose in the present Note to investigate in a 
general manner, and without entering into any analytical detail, 
the general form of the curve corresponding to different values 
of the quantity C. 
It is proper to remark, that the curve is not altered by 
changing the signs of each or any of quantities m, m’, C (in fact, 
analytically the distances r, 7! are essentially ambiguous in sign), 
so that we may without loss of generality consider m, m', C as 
all of them positive. The different branches of the complete 
analytical or geometrical curve have distinct mechanical signifi- 
! 
cancies ; thus r, 7 being positive, + a= C is the curve for 
‘which the potential of the attracting masses m, m! is equal to C ; 
mm ; : é 
but a “=O is the curve for which the attracting mass m, 
and the repulsive mass m', have the potential C ; but this is a 
distinction to which I do not attend. I write for homogeneity 
Oe : d : 
z instead of C, where a is the distance between the points M, M’; 
the equation thus becomes 
mm ok 
. iy Dis Ge 
Where a is a positive distance, m, m', k may be considered as 
positive abstract numbers. The curve is obviously a curve of 
the eighth order. When & is large in comparison with m, m’, 
then since r, 7’ cannot be both of them small in comparison of a 
(for if one be small, the other will be nearly equal to a), it is 
clear that one of these distances, for instance r, will be small, 
and the other 7’ nearly equal to a. We in fact have (neglecting 
! 
; ‘ m! . : Suoiiey) nen! Je 
in the first instance —- in comparison with —) —=-, or more 
i r / a 
m ktm. m f 
accurately, — = ———, i. e. r= -—,, which shows that a part 
r a k+m 
of the curve consists of two ovals, which are approximately con- 
centric circles, radii ie about the point M as centre. In 
like manner a part of the curve consists of two ovals, which are 
approximately concentric circles, radii a, about the point 
k+m 
M’ as centre. I denote by A, B, the two ovals about M, viz. A 
is the exterior, and B the interior oval; and in like manner by 
A’, B! the two ovals about M’, viz. A! is the exterior, and B! the 
