Hil [lilf| 
IStl 
A MEMOIR ON PREPOTENTIALS. 
Reverting to the original form of the two equations, and attending to the relation 
Q'+ Q" = 0, we obtain 
dW' dW" -4(ri) s+1 
+ -TT = “tVTTT—T\ p, 
or, what is the same thing, 
r (**-*) 
r (*«-*) fdW' 
11. I recall the signification of the symbols:—V', V" are the potentials, it may 
be different functions of the coordinates (a, .., c, e) of the attracted point, for positions 
of this point on the two sides of the surface (as to this more presently): and W', W" 
are what V', V" respectively become when the coordinates (a, .., c, e) are replaced by 
(x,..,z, w), the coordinates of a point N on the surface. The explanation of the 
dW' dW" 
symbols - is given a little above; p denotes the density at the point (x,.., z, w). 
12. The like remarks arise as with regard to the former distribution theorem (A); 
the functions V', V" cannot be assumed at pleasure; non constat that there is any 
distribution in space, and still less any distribution on the surface, which would give 
such values to the potential of a point (a, .., c, e) on the two sides of the surface 
respectively; but assuming that the functions V', V" are such that they do arise from 
a distribution on the surface, or say that they satisfy all the conditions, whatever they 
are, required in order that this may be so, then the formula determines the distri 
bution, viz. it gives the value of p, the density at a point {x, .., 2, w) of the surface. 
13. In the case where the surface is the plane w = 0, viz. in the case of the 
potential-plane integral, 
pdx... dz 
{(« — xf -f-... + (c — z) 2 + e 2 }4 s ~l 
(e assumed to be positive); then, since the conformation is symmetrical on the two sides 
of the plane, V' and V" are the same functions of (a, .., c, e), say they are each = V; 
W', W" are each of them the same function, say they are each = W, of (x, .., z, e) 
that V is of (a, .., c, e) ; the distribution-formula becomes 
r(*«-*) (dW\ 
p 2(ri)*«UeA.. 
viz. this is also what one of the prepotential-plane formulae becomes on writing therein