Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

A MEMOIR ON PREPOTENTIALS. 
331 
[607 
607] A MEMOIR ON PREPOTENTIALS. 331 
point {a,.., c, e), so long as this point in its course does not traverse the material 
mula has 
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 vary continuously as the point (a, .., c, e) passes continuously along 
the curve. 
erve that 
20. Consider a “region,” that is, a portion of spacfe 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. 
in com- 
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 
s +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 (s + 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 
prepotential 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 
1 for the 
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, s +1 = 3, of 
o proceed 
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 consider the circuit as corresponding to (in ordinary tridimensional space) a plane 
curve of equal periphery, the corresponding points on the circuit and the plane curve 
negative, 
being points at equal distances s along the curves from fixed points on the two 
curves respectively; and then treating the plane curve as the base of a cylinder, we 
may represent the potential as a length or ordinate, V = 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 prepotential curve, which represents the march of the prepotential. 
, . &c. ad 
all finite. 
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, 
ther cases 
then represents the value of the prepotential. The before-mentioned continuity-theorem 
the pre 
attracted 
is that the prepotential curve, corresponding to any portion (of the circuit) which 
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