PROFESSOR CAYLEY ON PREPOTENTIALS. 
679 
2. We now consider the prepotential-surface integral 
V= f 
J {{a-xf... + [c-zY 
y+{e-wyy s+q 
As already mentioned, it is only a particular case of this, the prepotential-plane integral, 
which is specially discussed ; but at present I consider the general case, for the purpose 
of establishing a theorem in relation thereto. The surface (s-dimensional surface) S is 
any given surface whatever. 
Let the attracted point P be situate indefinitely near to the surface, on the normal 
thereto at a point N, say the normal distance NP is=a* ; and let this point N be taken 
as the centre of an indefinitely small circular (s-dimensional) disk or segment (of the 
surface), the radius of which R, although indefinitely small, is indefinitely large in com- 
parison with the normal distance s. I proceed to determine the prepotential of the 
disk ; for this purpose, transforming to new axes, the origin being at N and the axes of 
x ... z in the tangent-plane at N, then the coordinates of the attracted point P will be 
(0 . . .0, »), and the expression for the prepotential of the disk will be 
V _ f q dx . . .dz 
J{A.. + ^+ 2 } is+s ’ 
where the limits are given by x 2 . . .-f-;s 2 <R 2 . 
Suppose for a moment that the density at the point N is =§', then the density 
throughout the disk may be taken =§', and the integral becomes 
dx . . .dz 
r =o' r — - 
where instead of g' I write g ; viz. g now denotes the density at the point N. Making 
this change, then (by what precedes) the value is 
2(ri) s r R r s ~'dr 
r (* s ) Jo {/• 2 + 8 2 } is+9 ' 
q= Positive. — Nos. 3 to 7. 
3. I consider first the case where q is positive. The value is here 
2(P1) S 1 J r^sTV/ f xi~'dx 
(1+tf) 5 ' 
or since is indefinitely small, the ^-integral may be neglected, and the value is 
_J_ (Tl)Tg 
_ 8 2 !?r(is+?)' 
Observe that this value is independent of R, and that the expression is thus the same 
as if (instead of the disk) we had taken the whole of the infinite tangent-plane, the 
* s is positive ; in afterwards writing s=0, we mean by 0 the limit of an indefinitely small positive quantity 
