332 
A MEMOIR ON PREPOTENTIALS. 
[607 
does not meet the material surface, is a continuous curve: viz. that there is no abrupt 
change of value either in the ordinate y (= V) of the prepotential curve, or in the 
first or any other of the derived functions &c. We have thus (in each of 
the two figures) a continuous curve as we pass (in the direction of the arrow) from 
a point P' on one side of the segment to a point P" on the other side of the 
segment; but this continuity does not exist in regard to the remaining part, from 
P" to P', of the prepotential curve corresponding to the portion (of the circuit) 
which traverses the material surface. 
22. I consider first the case q = — \ (see the left-hand figure): the prepotential 
is here a potential. At the point A, which corresponds to the passage through the 
material surface, then, as was seen, the ordinate y (= the Potential V) remains finite 
and continuous; but there is an abrupt change in the value of 
, that 
as 
is, in the 
direction of the curve: the point N is really a node with two branches crossing at 
this point, as shown in the figure; but the dotted continuations have only an analytical 
existence, and do not represent values of the potential. And by means of this branch- 
to-branch discontinuity at the point iY, we escape from the foregoing conclusion as to 
the continuity of the potential on the passage of the attracted point through a closed 
surface. 
23. To show how this is, I will for greater clearness examine the case (s+l) = 3, 
in ordinary tridimensional space, of the uniform spherical shell attracting according to 
the inverse square of the distance; instead of dividing the shell into hemispheres, I 
divide it by a plane into any two segments (see the figure, wherein A, B represent 
the centres of the two segments respectively, and where for graphical convenience the 
segment A is taken to be small). 
We may consider the attracted point as moving along the axis xx', viz. the two 
extremities may be regarded as meeting at infinity, or we may outside the sphere 
bend the line round, so as to produce a closed circuit. We are only concerned with 
what happens at the intersections with the spherical surface. The ordinates represent 
the potentials, viz. the curves are a, b, c for the segments A, B, and the whole 
spherical surface respectively. Practically, we construct the curves c, a, and deduce the 
curve b by taking for its ordinate the difference of the other two ordinates. The 
curve c is, as we know, a discontinuous curve, composed of a horizontal line and two 
hyperbolic branches; the curve a can be laid down approximately by treating the 
segment A as a plane circular disk; it is of the form shown in the figure, having 
a node at the point corresponding to A. (In the case where the segment A is