I 
144 Mr. A. Cayley on the Equipotential Curve “+ “=C. 
interior oval. The distances inter se of the ovals A and B, or of 
the ovals A! and B’, are small in comparison with the radii of these 
ovals respectively ; andif, to fix the ideas, m! be greater than M, 
then the ovals A’, B! are greater than the ovals A and B. 
It is easy to see that the curve will have a node or double 
point on the axis if kK=(/m! + “m)?; and we must first con- 
sider the case kK=( WV m+ Wm)2. The node lies between the 
points M, M!, and its distances from these points are respectively 
as /m: Vm, that is, it is nearest to M. The transition from 
the original form is very obvious; the exterior ovals A, A’ have 
gradually expanded until they come in contact, and at the 
instant of doing so the two ovals change themselves into a figure 
of eight, AA’. The ovals B, B! also expand and change their 
form, but they preserve the general character of ovals enclosing 
the points M, M! respectively. The curve consists of a figure of 
eight AA’, and (inside of the two divisions thereof respectively) 
of the ovals B, B! enclosing the points M, M’. The half of the 
curve nearest to M! is, as before, preponderant in magnitude. 
The next change when & continues to diminish is an obvious 
one: the figure of eight opens out into an hourglass-shaped 
oval AA!, while the ovals B, B! continue increasing in magnitude 
and altering their form. 
There will beagain anodeor double point whenk=( /m! — /m)?3 
but to explain the transition to this special form, it is necessary 
to attend more particularly to the change of form in the oval B! 
as k approaches to the value in question, viz. this oval lengthens 
out and begins to twist itself round the oval B; and when # 
becomes =( /m!— “m)?, then the oval B! has completely en- 
circled B, the two extremities of B' meeting together at the 
double point, which is a point beyond M (2. e. on the other side 
to M’), such that its distances from M, M! are in the ratio of 
/m:Vm'. And at the instant of contact there is, as in the 
former case, a modification of the form of the portions which 
come into contact, so that the node is an ordinary double point. 
The oval B! has, in fact, become what may be termed a re-entrant 
figure of eight, ©). the small part of which encloses the 
oval B which encloses the point M, while the large part encloses 
the point M’. The curve consists of the exterior oval AA’ (which 
has probably lost wholly or partially its hourglass form, and is 
more nearly an ordinary oval), of the re-entrant figure of eight, 
B’, and of the enclosed oval B. 
As k continues to diminish, the re-entrant figure of eight, B’, 
breaks up into two detached ovals /B!, mB’, the larger of which, 
1B!, encloses the other one and also the pomt M'; while the 
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