326 
A MEMOIR ON PREPOTENTIALS. 
[607 
where p here denotes the density at the point N; 
( 8 2 « 4 
= Jt(l + terms in 
and the value of the r-integral 
ryri 
8 T (f»-*)’ 
Observe that this is indefinitely small, and remains so for a point P on the surface; 
the potential of the remaining portion of the surface (for a point P near to or on 
the surface) is finite, that is, neither indefinitely large nor indefinitely small, and it 
varies continuously as the attracted point passes through the disk (or aperture in the 
material surface now under consideration); hence the potential of the whole surface 
is finite for an attracted point P on the surface, and it varies continuously as P 
passes through the surface. 
It will be noticed that there is in this case a term in V independent of a; 
and it is on this account necessary, instead of the potential, to consider its derived 
function in regard to a; viz. neglecting the indefinitely small terms which contain 
powers of I write 
dV _ 2(r^) s+1 
da r(T« — 
The corresponding term arising from the potential of the other portion of the 
surface, viz. the derived function of the potential in regard to a, is not indefinitely 
small; and calling it Q, the formula for the whole surface becomes 
dV n *(?&+' 
d« V r(Js-i) P ’ 
9. I consider positions of the point P on the two opposite sides of the point N, 
say at the normal distances a', a", these being positive distances measured in opposite 
directions from the point N. The function V, which represents the potential of the 
surface in regard to the point P, is or may be a different function of the coordinates 
(a, .., c, e) of the point P, according as the point is situate on the one side or the 
other of the surface (as to this more presently). I represent it in the one case by 
V', and in the other case by V"; and in further explanation state that a is measured 
into the space to which V refers, a" into that to which V" refers; and I say that 
the formulae belonging to the two positions of the point P are 
dW'_ 2(Tjy» 
do' V T(is-i) p ’ 
dV" 2(r j)-» 
do" W r (is-if’ 
where, instead of V', V", I have written W', W", to denote that the coordinates, as 
well of P' as of P", are taken to be the values (x, .., z, w) which belong to the 
point N. The symbols denote 
dW' 
dW' 
dW' 
da 
— —— cos a 
dx 
+ .. 
” + dz 
dW" 
dW" 
dW" 
da" 
= — ,— cos a 
dx 
+ ., 
dz