PROFESSOR CAYLEY ON PREPOTENTIALS. 
687 
Continuity of the Prepotential-surface Integral. — Art. Nos. 19 to 25. 
19. I again consider the prepotential-surface integral 
f & 
J {{a-xf . . . + {c-zf+ 
in regard to a point (a . . . c, e) not on the surface ; q is either positive or negative, as 
afterwards mentioned. 
The integral or prepotential and all its derived functions, first, second, &c. ad infinitum , 
in regard to each or all or any of the coordinates (a . . . c, e) are all finite. This is cer- 
tainly the case when the mass j %dS is finite, and possibly in other cases also ; but to fix 
the ideas we may assume that the mass is finite. And the prepotential and its derived 
functions vary continuously with the position of the attracted point (a . . . c, e), so long 
as this point in its course does not traverse the material surface. For greater clearness 
we may consider the point as moving along a continuous curve (one-dimensional locus), 
which curve, or the part of it under consideration, does not meet the surface ; and the 
meaning is that the prepotential and each of its derived functions varies continuously as 
the point (a . . . c, e) passes continuously along the curve. 
20. Consider a “ region,” that is, a portion of space any point of which can be by a 
continuous curve not meeting the material surface connected with any other point of 
the region. It is a legitimate inference, from what just precedes, that the prepotential 
is, for any point (a . . . c, e) whatever within the region, one and the same function of the 
coordinates {a . . . c, e), viz. the theorem, rightly understood, is true ; but the theorem 
gives rise to a difficulty, and needs explanation. 
Consider, for instance, a closed surface made up of two segments, the attracting 
matter being distributed in any manner over the whole surface (as a particular case 
5+1 = 3, a uniform spherical shell made up of two hemispheres) ; then, as regards the 
first segment (now taken as the material surface), there is no division into regions, but 
the whole of the (5+l)dimensional space is one region; wherefore the prepotential 
of the first segment is one and the same function of the coordinates (a . . . c, e) of the 
attracted point for any position whatever of this point. But in like manner the prepo- 
tential of the second segment is one and the same function of the coordinates (a . . . c, e) 
for any position whatever of the attracted point. And the prepotential of the whole 
surface, being the sum of the prepotentials of the two segments, is consequently one and 
the same function of the coordinates (a . . . c, e) of the attracted point for any position 
whatever of this point ; viz. it is the same function for a point in the region inside the 
closed surface and for a point in the outside region. That this is not in general the case 
we know from the particular case, 5 + 1 = 3, of a uniform spherical shell referred to above. 
21. Consider in general an unclosed surface or segment, with matter distributed over 
it in any manner ; and imagine a closed curve or circuit cutting the segment once ; and 
let the attracted point (a. .. c,e) move continuously along the circuit. We may con- 
sider the circuit as corresponding to (in ordinary tridimensional space) a plane curve of 
