688 
PROEESSOK CAYLEY ON PREPOTENTIALS. 
equal periphery, the corresponding points on the circuit and the plane curve being 
points at equal distances s along the curves from fixed points on the two curves respec- 
tively ; and then treating the plane curve as the base of a cylinder, we may represent 
the potential as a length or ordinate, Y=y, measured upwards from the point on the 
plane curve along the generating line of the cylinder, in such wise that the upper 
extremity of the length or ordinate y traces out on the cylinder a curve, say the prepo- 
tential curve, which represents the march of the prepotential. The attracted point may, 
for greater convenience, be represented as a point on the prepotential curve, viz. by the 
upper instead of the lower extremity of the length or ordinate y ; and the ordinate, or 
height of this point above the base of the cylinder, then represents the value of the 
prepotential. The before-mentioned continuity-theorem is that the prepotential curve 
corresponding to any portion (of the circuit) which 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 
dy d^y 
&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^=— ^ (see the left-hand figure): the prepotential is 
here a potential. At the point N, wdiich 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 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 N, 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 
