An introduction to the mathematical theory of attraction ...

Differential Coefficients of the Potential. 75
up of two parts, of which one is due to a circular plate, of
infinitely small radius a, having O for centre, and the other
to the mass whose distance from O exceeds a.
Taking O for origin, the normal at O for the axis of z,
and putting x2 + y2 = p2, we have
-f f ap cos 6 dp d { a x d
X = + - dxdy,
JoJo p2 JJCOS r3
where 4 denotes the angle which the normal to the surface
at any point makes with the axis of z.
In the first integral
(da\ fd(\
= U-0 + -) p) cos + }p sin.
Hence the first integral is of the order a. In the second
integral, z is a function of æ and y, given by the equation of
the surface; whence
d /lf idr dr dz
dx \r) r2 \dx dz dx
therefore, denoting the second integral by X, and integrating
by parts, we have
r cos4 JJ\dx kcos 4 cos r2 d d dy
The limiting curve next the origin, round which the
single integral is to be taken, is a circle of radius a, at whose
circumference r and cos 4 differ from a and unity respectively
by infinitely small quantities of the second order; also
dy = a cos 4 dà, and c differs from a constant by a quantity
of the order a. Hence for this curve ' 4 = O.
J r cos +
Inside the sign of double integration dx dy = p dp dp, and
when r is small, z is of the order r2, and therefore the coefficient of dp df is always finite. Hence we conclude that X
is finite.