262 
ON A SMITH’S PRIZE QUESTION, RELATING TO POTENTIALS. 
[757 
A is based upon the consideration of this segment alone; and, on the ground that 
we can without crossing the segment pass from any one position of P to any other 
position of P, it is inferred that the potential is one and the same function of the 
coordinates, whatever be the position of P: it is therefore unassailable by any 
considerations in relation to the non-existent segment B. Similarly the assertion in 
regard to the potential of the segment B is based upon the consideration of this 
segment alone, and it is unassailable upon any considerations in regard to the non 
existent segment A: the potential of the whole sphere is certainly the sum of the 
potentials of the segments A and B: it is therefore altogether off the purpose to 
object that in the case of the whole sphere we cannot pass from a point outside 
the sphere to a point inside the sphere without crossing one or other of the segments 
A and B. I consider that the two assertions are each of them true, and that the 
conclusion is a legitimate one, but it is true only in the sense in which a + x + */[(a — a-) 2 ] 
is one and the same function of x whatever be the value of x: this is so, if 
\J[(a — xY] denotes indifferently or successively the two functions + (a — x): but if, a 
and x being real, — x)*] is taken to mean the positive value, then the function 
a + x + V[( a — X Y] is = 2a or = 2x according as a — x is positive or negative. 
Fig. l. 
In further illustration, let the dark line of fig. 1 represent the intersection of 
an unclosed surface, or segment, by the plane of xz taken to be that of the paper, 
and consider the potential of the segment in regard to a point P in the plane of 
the paper, coordinates x, z. We have the potential V defined as a function of x, z 
by an equation V= a definite integral, depending on the parameters x, z, and being in 
general a transcendental function of (x, z)\ V is a real, one-valued, finite, continuous 
function of x, z: in particular, if the point P, moving in any manner, traverses the 
dark line, there is not any discontinuity in the value of V. There is however in 
this case a discontinuity in the differential coefficients of F: if to fix the ideas we 
imagine P moving parallel to the axis of x, so that 2 is taken to be constant and 
V a function of x only, then when the path of P crosses the black line there is 
dV 
in general an abrupt change of value in . Taking V as a coordinate y at right 
angles to the plane of the paper, a section by any plane parallel to that of xy is 
(when the trace of the plane upon that of xz does not meet the dark line) a 
continuous curve; but when the trace meets the dark line, then for this value of x 
there is an abrupt change of direction in the section.