690 
PROFESSOR CATLET ON PREPOTENTIALS. 
24. In the case q positive the prepotential curve is as shown by the right-hand 
figure in p. 688, 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 
-j- 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 on an indefinitely 
near point is found as before, and the prepotential of the whole surface is 
_i JI TO, V 
n 
where V 15 the prepotential of the remaining portion of the surface, is a function which 
varies (and its derived functions vary) continuously as the attracted point traverses the 
disk. To fix the ideas we may take the origin at the centre of the disk, and the axis 
of e as coinciding with the normal, so that s, which is always positive, is =+e; and 
the expression for the prepotential at a point (a ... c, e) on the normal through the 
centre of the disk is 
, (rp*.r g v 
( ±e )2 2 -f r(i s +g') i " 1 
viz. when q is fractional there is the discontinuity of law, inasmuch as the term changes 
from 
( + <?) 2 
to 
i-ey 
but when q is integral this discontinuity disappears. The like 
considerations, using of course the proper formula for the attraction of the disk, would 
apply to the case q=0 or negative. 
25. Or again, we might use the formulae which belong to the case of a uniform (s+ 1)- 
coordinal spherical shell (see Annex No. III.), viz. we decompose the surface as follows, 
surface = disk residue of surface; 
and then, considering a spherical shell touching the surface at the point in question 
(so that the disk is in fact an element common to the surface and the spherical shell), 
and being of a uniform density equal to that of the disk, we have 
disk = spherical shell— residue of spherical shell ; 
and consequently 
surface = spherical shell — residue of spherical shell+residue of surface; 
and then, considering the attracted point as passing through the disk, it does not pass 
through either of the two residues, and there is not any discontinuity, as regards the 
prepotentials of these residues respectively ; there is consequently, as regards the pre- 
potential of the surface, the same discontinuity that there is as regards the prepotential 
of the spherical shell. But I do not further consider the question from this point of view. 
