605] 
NOTE ON A POINT IN THE THEORY OF ATTRACTION. 
313 
shell ; for the potential is a different function for external and interior points, viz. 
. . M 
for internal points it is a constant, = iff 4- radius ; for external points it is = 
Va 2 + b 2 + c 2 ’ 
if a, by c are the coordinates measured from the centre of the sphere. 
The difficulty is rather apparent than real. Reverting to the case of an unclosed 
surface or segment, and considering the continuous curve from P to Q, let this be 
completed by a curve from Q to P through the segment; viz. we thus have P, Q 
points on a closed curve or circuit meeting the segment in a single point. To fix 
the ideas, the circuit may be taken to be a plane curve, and the position of a point 
on the circuit may be determined by means of its distance s from a fixed point on 
the circuit. Considering this circuit as drawn on a cylinder, we may at each point 
of the circuit measure off, say upwards, along the generating line of the cylinder, a 
length or ordinate z, proportional to the potential of the point on the circuit, the 
extremities of these distances forming a curve on the cylinder, say the potential curve. 
We may draw a figure representing this curve only; the points P, Q being marked 
as if they were points on the curve (viz. at the upper instead of the lower extremities 
of the corresponding ordinates z): the generating lines of the cylinder, and the plane 
section which is the circuit, not being shown in the figure. The potential curve is 
then, as shown in the figure, a continuous curve, viz. we pass from P to Q in the 
direction of the arrow, or along that part of the circuit which does not meet the 
segment, a curve without any abrupt change in the value of the ordinate z or of 
dz d^z 
any of its differential coefficients, ^^, &c.; but there is, corresponding to the 
point where the circuit meets the surface, an abrupt change in the direction of the 
dz 
potential curve or value of the differential coefficient viz. the point on the curve 
is really a node, the two branches crossing at an angle, as shown by the dotted 
lines, but without any potentials corresponding to these dotted lines. 
In the case of two segments forming a closed surface, or say two segments forming 
a complete spherical shell; then, if the points P, Q are one of them internal, the other 
external, the circuit, assuming it to meet the first segment in one point only, will meet 
the second segment in at least one point; the potential curves corresponding to the two 
segments respectively will have each of them, at the point corresponding to the intersec 
tion of the circuit with the segment, a node; and it hence appears how, in the potential 
curve corresponding to the whole shell (for which curve the ordinate 2 is the sum 
of the ordinates belonging to the two segments respectively), there will be a dis 
continuity of form corresponding to the passage from an exterior to an interior point. 
C. IX. 40