11] 



FIGURE OF THE EARTH. 



205 



The value of the integral in this expression may be written down in 

 several cases. 



Ex. 1 . f(p}= 1 1 p. Integral = 2ar (r 2 + a 2 ) - (r 2 - a 2 ) 2 log,. - - . 



r a 



2. f(p) = l/p\ Integral = ~ a 3 . 



o 



3. f(p) = l/p 3 . Integral = - 4ra + 2 (r + cr) log e - - . 



V "~~ (/ 



4. f(p) = l/p\ Integral =-^- s . 







3. Comparison of the attractions of a solid sphere upon any external 

 point and a related internal point. 



Let M be the external point and O the centre of the sphere. In OM 

 take M' so that OM, OA, OM' are in continued proportion; thus if 



OM' = r', r' = a*/r. 



Let Q be any point of the circle with centre and radius OA, and 

 let Qq be an indefinitely small arc of this circle. Join MQ, M'Q. 



Then since M'O : OQ = OQ : OM, 



the triangles M'OQ, QOM are similar, and we have 



M'Q :MQ = OQ:OM; 

 so that M'Q is to MQ in the constant ratio OQ : OM or a : r. 



Let shells be described in the sphere with centres M, M' and internal 

 and external radii respectively MQ, Mq, M'Q, M'q. Call MQ, Mq, p, p + 8p 

 respectively, and M'Q, M'q, p', p' + Bp'. Then if we take the attraction of 

 a particle to vary inversely as the nth power of the distance, we have 



Attraction on M of shell with centre M = 8p . QS\ 



