PROFESSOR CAYLEY ON PREPOTENTIALS. 
691 
The Potential Solid Integral. — Art. No. 26. 
26. We have farther to consider the prepotential (and in particular the potential) of 
a material space ; to fix the ideas, consider for the moment the case of a distribution 
over the space included within a closed surface, the exterior density being zero, and the 
interior density being, suppose for the moment, constant ; we consider the discontinuity 
which takes place as the attracting point passes from the exterior space through the 
hounding surface into the interior material space. We may imagine the interior space 
divided into indefinitely thin shells by a series of closed surfaces similar, if we please, 
to the bounding surface ; and we may conceive the matter included between any two 
consecutive surfaces as concentrated on the exterior of the two surfaces, so as to give 
rise to a series of consecutive material surfaces ; the quantity of such matter is infini- 
tesimal, and the density of each of the material surfaces is therefore also infinitesimal. 
As the attracted point comes from the external space to pass through the first of the 
material surfaces — suppose, to fix the ideas, it moves continuously along a curve the 
arc of which measured from a fixed point is =s — there is in the value of V (or, as the 
dV 
case may be, in the values of its derived functions &c.) the discontinuity due to the 
passage through the material surface ; and the like as the attracted point passes 
through the different material surfaces respectively. Take the case of a potential, 
q— — ^ ; then, if the surface-density were finite, there would be no finite change in the 
dV 
value of Y, but there would be a finite change in the value of ; as it is, the changes 
are to be multiplied by the infinitesimal density, say g, of the material surface ; there is 
consequently no finite change in the value of the first derived function ; but there is, 
or may be, a finite change in the value of and the higher derived functions. But 
there is in V an infinitesimal change corresponding to the passage through the successive 
material surfaces respectively ; that is, as the attracted point enters into the material 
space there is a change in the law of V considered as a function of the coordinates 
(a ... c, e) of the attracted point ; but by what precedes this change of law takes place 
without any abrupt change of value either of V or of its first derived function ; which 
derived function may be considered as representing the derived function in regard to 
any one of the coordinates a . . ,c,e. The suppositions that the density outside the 
bounding surface was zero and inside it constant, were made for simplicity only, and 
were not essential ; it is enough if the density, changing abruptly at the bounding 
surface, varies continuously in the material space within the bounding surface*. The 
* It is, indeed, enough if the density varies continuously within the hounding surface in the neighbourhood, 
of the point of passage through the surface ; but the condition may without loss of generality be stated as in 
the text, it being understood that for each abrupt change of density within the bounding surface we must 
consider the attracted point as passing through a new bounding surface, and have regard to the resulting- 
discontinuity. 
4 z 
MDCCCLXXV. 
