An introduction to the mathematical theory of attraction ...

140 The Potential.
the direction angles of this line be denoted by a, 3, y, then
we have
x }/ y
- COS a, =COS, - =os y,
r r r
also Jx'2dm = f r'2dm - A, with two similar equations; and
neglecting terms of an order higher than (1 ), we get
V=- + 23 {3 (x'2 cos2 a + y'2 c2 C 3 + zt2 COS2 Y) - r'2) dm
M 1
= + 23 { 2 fr'2dm - 3(A cos2 a +B cos2 3 + C cos2 7)}
- M 1
-- + (r + B + C-3 ). (2)
It follows from equation (1) that, at a point P so distant
t rI2 )
from M that (- is negligible, the potential of Mis the same
as if the entire mass were concentrated at its centre of inertia.
The term moment of inertia when applied to electric mass
signifies merely an integral depending on the positions and
intensities of the force-centres of which the electric mass is
composed. A remark of a similar character applies to the
term centre of inertia.
79. Moment exerted by Distant Body.-If M be a
rigid body, and a mass L be concentrated at a distant point
P, remembering that in the case of mutually attracting
bodies the potential is a force function, we see that the
moments round the axes of the force which M exerts on L are
L ( - dV) &c.;
( dyv idx
but since these are equal and opposite to the moments
exerted by L on M, if these latter be denoted by Ni, N2, N3,
we have
N, =L/ dV d V
-lx dy &