607] 
A MEMOIR ON PREPOTENTIALS. 
333 
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 
n 
A 
x 
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 c, 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 consequence 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. 
24. In the case q positive, the prepotential curve is as shown by the right-hand 
figure on p. 332, viz. the ordinate is here infinite at the point N corresponding to 
the passage through the surface; the value of the derived function changes between 
+ infinity and — infinity; and there is thus a discontinuity of value in the derived 
function. It would seem that, when q is fractional, this occasions a change of law 
on passage through the surface: but that there is no change of law when q is 
integral. 
In illustration, consider the closed surface as made up of an infinitesimal circular 
disk, as before, and of a residual portion; the potential of the disk at an indefinitely 
near point is found as before, and the prepotential of the whole surface is