acted on by small Forces. 205 
point: x, y, 2, the co-ordinates of any point: let the forces be 
considered positive when they tend to make a, 5, c, increase: call 
U’, X’, ¥’, Z, those parts of U, X, ¥, Z which arise from these 
attractions. 
Pheri poe ae : ; 
dx dy dz VY \(w—a)* + (y—b)? + (z—c)*}’ 
F av v-—a 
**dadxdydz~ {(«—a) + (y—b)? + (z-c)*}3 
de X’ dill Bae wobe lay is 
= da dydv °"dxdydzda’~ dxdydz’ °‘da~~° 
dV “BF : : , 
= Z,, (the limits of integration being supposed 
Similarly a au. 
independent of a, b, c,) and the value of U’ is therefore V. If 
then we take the sum of the products of each attracting particle 
into the reciprocal of its distance from the point whose co-ordinates 
are a, b, c, and if we add that part of U which originates from 
the extraneous forces, and make the sum =C, we shall have 
the equation to the surface. 
(3.) Our object then, at present, is to find the value of 
this sum for a point in the surface; and we might suppose the 
disturbing forces to be any whatever. But in nearly all the 
cases to which we can apply this investigation, the forces act 
symmetrically round an axis. Suppose, then, that the forces are 
symmetrical about an axis: the body will then be a solid of 
revolution; and, for the sake of simplicity, we will make b=o0. 
We proceed to investigate that part of Y which arises from the 
attraction of the particles of the body. 
(4.) First, for a small pyramid whose vertex is the attracted 
point. Let p be its whole length, p any variable length, 4 the 
area of a section perpendicular to its axis at distance 1 from the 
vertex: the area at distance p is Ap*®; the sum of the products 
of the masses into the reciprocals of their distances, for the slice 
included between the lengths p and p+ dp is ultimately Apép; 
