investiga

(a) For the analytical proof of these propositions, consider, first, Analytical a pair of particles, O and P, whose masses are m and unity, and tion of the co-ordinates a, b, c, x, y, z. If D be their distance

and therefore the work required to remove P to infinity is

value of the potential.

which, since the superior limit is D =∞, is equal to

The mutual potential energy is therefore, in this case, the product of the masses divided by their mutual distance; and there

fore the potential at x, y, z, due to m, is

m

D

Again, if there be more than one fixed particle m, the same investigation shows us that the potential at x, y, z is

m

D'

And if the particles form a continuous mass, whose density at a, b, c is p, we have of course for the potential the expression

dadbdc
D

J S S p dad

the limits depending on the form of the mass.

any point.

If we call the potential at any point P (x, y, z), it is evi- Force at dent (from the way in which we have obtained its value) that the components of the attraction on unit of matter at P are

All this is evidently independent of the question whether P lies within the attracting mass or not.

Force within a homo

geneous sphere.

Rate of increase of the force in any direction.

(b) If the attracting mass be a sphere of density p, and centre a, b, c, and if P be within its surface, we have, since the exterior shell has no effect,

1

4

3

D3

== πρ.

d2 d2 d2

V2= + +
dx2 + dy2 dz2'

we have 2
v2=0, as was proved before, App. B (g), (14). For

Laplace's

equation.

and from this, and the similar expressions for the second differentials in y and 2, the theorem follows at once.

and
ρ does not involve x, y, z, we see that as long as D does not
vanish within the limits of integration, t.e., as long as P is not a
point of the attracting mass

dx dy dz
+
dx dy

+ dz

=0.

If P be within the attracting mass, suppose a small sphere to be described so as to contain P. Divide the potential into two parts, V1 that of the sphere, V, that of the rest of the body. The expression above shows that

Poisson's ex

tension of

Laplace's

equation.

which is the general equation of the potential, and includes the

extension

case of P being wholly external to the attracting mass, since Poisson's then p=0. In terms of the components of the force, this equation of Laplace's becomes

dz

dx dy
+ +
dx dy dz

-=

Απρ.

(d) We have already, in these most important equations, the means of verifying various former results, and also of adding new

ones.

equation.

of matter

concentric

Thus, to find the attraction of a hollow sphere composed of Potential concentric shells, each of uniform density, on an external point arranged in (by which we mean a point not part of the mass). In this case spherical symmetry shows that V must depend upon the distance from shells of the centre of the sphere alone. Let the centre of the sphere be density. origin, and let

r2 = x2+y2+z2.

Then is a function of r alone, and consequently

uniform

Hence, when P is outside the sphere, or in the hollow space within it,

A first integral of this is

2 dV d'V

+ =0.
dr.s

=C. dr

r dr
redv

For a point outside the shell C has a finite value, which is easily seen to be M, where M is the mass of the shell.

For a point in the internal cavity C=0, because evidently at

dV the centre there is no attraction-i.e; there r=0, =0 together. dr

Hence there is no attraction on any point in the cavity.

We need not be surprised at the apparent discontinuity of this solution. It is owing to the discontinuity of the given distribution of matter. Thus it appears, by § 491 (c), that the true general equation to the potential is not what we have taken above, but d2 V 2 dv + dra r dr

where p, the density of the matter at distance r from the centre, is zero when r<a the radius of the cavity: has a finite value σ, which for simplicity we may consider constant, when r>a and <a' the radius of the outer bounding surface: and is zero, again,

if M, denote the whole amount of matter within the spherical surface of radius r; which is the discontinuous function of r specified as follows:

We have entered thus into detail in this case, because such apparent anomalies are very common in the analytical solution of physical questions. To make this still more clear, we subjoin a graphic representation of the values of V, and

dv
dr

ďV dr

for this case
and has a point of inflection at Q: but there is no discontinuity

ABQC, the curve for V, is partly a straight line,

dr.2

and no abrupt change of direction. OEFD, that for
continuous, but its direction twice changes abruptly. That for
d' V
consists of three detached portions, OE, GH, KL.

dv dr

is

B

E

F

cylinders of

(e) For a mass disposed in infinitely long concentric cylindrical Coaxal right shells, each of uniform density, if the axis of the cylinders be z, uniform we must evidently have V a function of x2+y2 only.

dv

Hence =0, or the attraction is wholly perpendicular to the

axis.

dz

density and infinite length.

from which conclusions similar to the above may be drawn.

(f) If, finally, the mass be arranged in infinite parallel planes, each of uniform density, and perpendicular to the axis of x; the resultant force must be parallel to this direction that is to say, Y=0, Z=0, and therefore

dX
dx

which, if p is known in terms of x, is completely integrable. Outside the mass, p=0, and therefore

X=C,

or the attraction is the same at all distances-a result easily verified by the direct methods.

If the mass consist of an infinite plane lamina of thickness t, and constant density p; then, supposing the origin to be half-way between its faces,

Matter ar

ranged in infinite parallel planes of uniform

density.

Outside the lamina X=C, (since p=0). At the positive surface,
and everywhere beyond it, C1=-2πpt, and at and beyond the
negative it is +2mpt. The difference of these is -4πpt (§ 478).
dV
is the attraction resolved in the Equi-
ds

(g) Since in any case
direction of the tangent to the arc s, it will be wholly perpen-
dicular to that arc if

potential surface.