An introduction to the mathematical theory of attraction ...

Discontinuity of Force in Surface Distribution. 13
quantity 8 of the second order the points of the surface whose
distances from P are less than an infinitely small quantity a
of the first order lie inside a sphere whose equation referred
to O as origin is x2 + y2 + (z - S)2 = a2, the normal at O being
taken as the axis of z; aiso the equation of the surface is of
the form z = t, + &c., and the points of the surface whose
distances from P are of the first order lie on the surface
z = u,, where u2 is a quadratic function of x and y. The
equation of the projection on the tangent plane at O of the
curve of intersection of this surface and the sphere is
2 2
2 + y2 + (U2 - )2= a2.
Infinitely small quantities of an order higher than a2 being
neglected, this becomes x2 + y2 = a2, which represents a circle of
radius a.
The projection on the tangent plane at O of an element
of surface dS is dS cos b, where 4p is the angle which the
normal to the element dS makes with OP. When dS and
P are infinitely near, cos h differs from unity by an infinitely
small quantity of the second order, also the distances from P
of the surface element and its projection differ by a quantity
infinitely small as compared with these distances, and finally
the difference of the angles which these distances make with
PO is infinitely small compared with the angles themselves.
Hence, if their densities be the same, the force exerted at
P by the elements of the surface S at a distance from P
less than a is the same as that due to their projections on the
tangent plane at 0, that is, the same as that due to the circular plate whose centre is O, whose radius is a, and whose
density is a the surface density at O.
We conclude therefore, by Art. 14, that the surface mass
infinitely near P exercises no attraction at P in the direction
of a tangent at 0, but produces a force in the direction of the
normal whose magnitude is expressed by 27ro. If the mass
be repulsive, this force is away from the surface on whichever
side of it P be situated.
It appears then from what has been said that as P
approaches and passes through the surface S at the point O
the tangential components at P of the repulsion of the mass