322 
A MEMOIR ON PREPOTENTIALS. 
[607 
value is required in the case where the distance e (taken to be always positive) is 
indefinitely small in regard to the radius R. 
Writing x = r%,.., z = r%, where the « new variables £, .., £ are such that I 8 + ... + £ 2 = 1, 
the integral becomes 
where dS is the element of surface of the s-dimensional unit-sphere + ... -f £ 2 = 1; the 
Jo (r 2 + e 2 )i s+ v’ 
is the r-integral of Annex II. 
2. We now consider the prepotential-surface integral 
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 («-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 = «*; and let this point 
N be taken at the centre of an indefinitely small circular («-dimensional) disk or 
segment (of the surface), the radius of which R, although indefinitely small, is in 
definitely large in comparison with the normal distance «. 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 
where the limits are given by x 2 + ... + z 2 < R 2 . 
Suppose for a moment that the density at the point N is = p, then the density 
throughout the disk may be taken = p', and the integral becomes 
where instead of p I write p; viz. p now denotes the density at the point N. 
Making this change, then (by what precedes) the value is 
2 <r& f R r 8 1 dr 
~ P TV1A I. 
T(i«) Jo {r 2 + « 2 p+9' 
* « is positive; in afterwards writing 8=0, we mean by 0 the limit of an indefinitely small positive 
quantity.