ATTRACTION OF SPHERICAL BODIES.

139

This result shews that the shell attracts the particle at C

in the same manner as if the mass of the shell were condensedinto its centre.

149- It follows also that a sphere which is either homo-

geneous or consists of concentric spherical shells of uniformdensity will attract the particle at C in the same manner asif the whole mass were collected at its centre.

Prop. To find the attraction of a homogeneous sphericalshell of small thickness on a particle placed within it.

150. We must proceed as in the last Proposition: butthe limits of y are in this case r — c and r + c : hence

prdr r r ~ e f r 2 — c s '

attraction of shell =

dy

r + c

(2c — 2c) = 0

therefore a particle within the shell is equally attracted in every

direction.

Prop. To find the attraction of a homogeneous sphericalshell on a particle without it; the law of attraction being re-presented by <p (y), y being the distance.

151. The calculation is exactly analogous to that ofArt. 148 : we have only to alter the law of attraction: thenattraction on C in CO

+ c 2 — r s ) <p (y) dy, (integrated by parts)

= {(/+ c 8 - r 1 ) f(p (y) dy - 2 f[yf<t> (y) dy] dy}

= !(?/+ r ~) <p\ (it) ~ 2x l r (y) + const.} suppose

between the specified limits

<pi(c-r)+-j^(c-r)

2wprdr

fiH c + r )

d (\js (c + r) — \js (c — r)

Zirprdr