mass at a point P is to be found by adding the quotients of every Potential portion of the mass, each divided by its distance from P.

due to an attracting point.

investiga

a. For the analytical proof of these propositions, consider, Analytical first, a pair of particles, O and P, whose masses are m and unity, tion of the and co-ordinates abc, xyz. If D be their distance

D2 = ( x − a)2 + (y − b)2 + (≈ − c)2.

The components of the mutual attraction are

value of the potential.

and therefore the work required to remove P to infinity is

'(x − a) dx + (y — b) dy + (z − c) dz

m

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

m

D'

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

the limits depending on the boundaries of the mass.

any point.

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

Force at any point.

Hence the force, resolved along any curve of which s is the arc,

Force within a homo

geneous sphere.

dV
ds

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

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

we have v

D

=0, as was proved before, App. B g (14) as a

particular case of g. The proof for this case alone is as follows:

and from this, and the similar expressions for the second differentials in y and 24, the theorem follows by summation.

Laplace's equation.

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

If P be within the attracting mass, suppose a small sphere Laplace's equation. to be described so as to contain P. Divide the potential into two parts, V, that of the sphere, V, that of the rest of the body.

1

The expression above shows that

2

which is the general equation of the potential, and includes the case of P being wholly external to the attracting mass, since there p=0. In terms of the components of the force, this equation becomes

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

new ones.

Poisson's extension of Laplace's

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 I must depend upon the distance from uniform the centre of the sphere alone. Let the centre of the sphere be origin, and let

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

shells of

density.

Potential of matter

arranged in concentric spherical shells of uniform density.

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 of the potential is not what we have taken above, but d'V 2 dV + dr2 r dr

Απρ,

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, for all values of r exceeding a'. Hence, integrating from r = 0,

dV

=

to r=r, any value, we have (since r2 O when r =

dr

[ pr2 dr = - M1,

(0),

if M, denote the whole amount of matter within the spherical surface of radius ; 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 subdv

join a graphic representation of the values of V,

dr

d3 V

and

dra

for this case. ABQC, the curve for V, is partly a straight line, and has a point of inflection at Q: but there is no discontinuity

Y

and no abrupt change of direction. OEFD, that for

continuous, but its direction twice changes abruptly. day

dr2

consists of three detached portions, OE, GA, KL.

Potential is of matter arranged in concentric

That for spherical

shells of uniform density.

For a mass disposed in infinitely long concentric cylin- Coaxal right drical shells, each of uniform density, if the axis of the cylinders

be z, we must evidently have V a function of x2 + y2 only.

0, or the attraction is wholly perpendicular to the

0; and therefore by (d)

cylinders of uniform density and infinite length.

Hence

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