208 FIGURE OF THE EARTH. [11 



If f(p) = , the expression for the attraction of the circular lamina is 



2-rra 1 - - . . 

 (cr + r 2 )-J 



To deduce the attraction of a solid of revolution divide it into plates 

 by planes perpendicular to its axis. The attraction of any plate is found 

 as above, and the sum of the attractions for all the plates gives the 

 attraction of the whole solid. 



1. Let the body be a cylinder of radius a; let b, c be the distances 

 of the attracted point from its two ends. Then if f(p) = l/p\ the attraction 

 of the whole cylinder is 



27r [c - b - (c 2 + a 2 )* + (V + ')*] 



2. Let the body be a spheroid, its equatorial and polar axes being 

 respectively a and c, and the distance of the attracted point from the 

 centre being r ; the attraction of a slice at distance x from the centre is 



r+g dx. 



The integral is to be taken from x = c to x = c ; its expression requires 

 circular or logarithmic functions according as c is less or greater than a ; 

 in the former case, the oblate spheroid, we find the attraction 



2 f^rc _nrc 

 \a--c- (<f-cf 



re + a" c 2 re a? + c 2 



where sin <t>= j. , sin <p = j= . 



a Jr + cf - <? a Jr* + a 2 - c 2 



It may be verified that this agrees with the result of Newton's 

 Prop. xci. Cor. 2, when his conic KRM is an ellipse. 



6. Attraction and potential of a spheroid differing little from a 

 sphere, upon a point situated upon the axis of revolution. 



Let M be the point ; draw MP intersecting the inscribed sphere in 

 P ; let MP p, and take p as independent variable ; draw Mp a con- 

 secutive, P'PN perpendicular to the axis, OPL passing through O the 

 centre, OK, pq perpendicular to MP. Let OP = c, OM=r. 



The surface of the elementary zone generated by revolution of Pp about 

 the axis is 2w . PN . Pp ; 



but Pp :Pq ::OP: OK, 



and PN : OK :: MP : MO ; 



