ATTRACTION OF SPHERICAL BODIES.

141

= 27t prdr — \ - 2 + 2 Jr}

* W /» '

dc

= 0, (see Art. 150.)

Ex. 2. Let (p (f) = r;

+ J, \js(r)

r 2 + B.

Attraction on an external particle

d f(c + r) 4 - (c - r) 4 + 4 A { (c + r) 2 - (c - r) 2 }

2 t rprdr — {- -------—-'-1- LI

r dc\ 8c

= 2 Trprdr— {eV + r 3 + 2 .dr}

- 47rpr 2 drc = mass of shell x c.

The attraction is the same as if the shell were collected at itscentre. This property we discovered for the law of the inversesquare. We shall now ascertain whether there are any otherlaws which give the same property.

Prop. To find what laws of attraction allow us tosuppose a spherical shell condensed into its centre whenattracting an external particle.

153. Let <p (r) he the law of force: then if c be thedistance of the centre of the shell from the attracted point andr the radius of the shell, and yfs(r) = f{rf(p (r) dr} dr, thenthe attraction of the shell

j^Kc + r) - (c - r) j

d

= 2t rprdr —

But if the shell be condensed into its centre this attraction

2r t > ( c ) = jj.

= 47 rr 2 drp(p(c) ;d f\], (c + r) -

'!' ( e

c