PROFESSOR CAYLEY ON PREPOTENTIALS. 689 
centres of the two segments respectively, and where for graphical convenience the seg- 
ment A is taken to be small. 
We may consider the attracted point as moving along the axis ocx', 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 corre 
sponding to A. [In the case where the segment A is actually a plane disk, the curve 
is made up of portions of branches of two hyperbolas ; but taking the segment A as 
being what it is, the segment of a spherical surface, the curve is a single curve, having 
a node as mentioned above.] And from the curves c and a, deducing the curve b, we 
see that this is a curve without any discontinuity corresponding to the passage of the 
attracted point through A (but with an abrupt change of direction or node corresponding 
to the passage through B). And conversely, using the curves a, b to determine the 
curve <?, we see how, on the passage of the attracted point at A into the interior of the 
sphere, in consequence of the branch-to-branch discontinuity of the curve a, the curve 
c, obtained by combination of the two curves, undergoes a change of law, passing 
abruptly from a hyperbolic to a rectilinear form, and how similarly on the passage 
of the attracted point at B from the interior to the exterior of the sphere, in conse- 
quence of the branch-to-branch discontinuity of the curve b, the curve c again 
undergoes a change of law, abruptly reverting to the hyperbolic form. 
