258 Mr. Sharpe on the Solid of greatest Attraction. 



5. The attraction of the solid to the point A is the fluent of 



t a s 



2p\x = 2px — 6 P a — x which when x becomes 



4 p a 

 a, is = - 



5 



6. And since the attraction of a sphere to a point on its 



surface = ■ | , which with equal contents = -£- X — ^~ 

 the attraction of this solid to that of a sphere is as 1026 : 1000. 



7. Again, since the attraction of this solid = — ^-, and of 



a sphere ~ — if we make their attraction equal — = a -— . 



8. Let d be the distance from A of a point on the axis, 

 into which the whole solid might be concentrated without al- 

 tering its attraction on A, then the attraction which = — ^— 



also =3 —— - x — r , and d = a -r^t which in a sphere s= 



radius = a 



1000 



9. The distance of its centre of gravity from A is = 



4 8 4 



fluent of \py*xx 1« t i t -|j . , . . 468 



75 — r — ph — 5-t" = — 2 — 1 = (when x is = a) a — — . 



fluent of \ptfx 3 4 -/- , 3 v ' 1000 



10. The ordinate is a maximum when the fluxion of y~ = 



2xx: hence x = a 439 when y is a maximum. 



3** 100 ° 



11. Hence we obtain the following very curious points 

 on the axis of the solid (fig. I.), with their distances from A 

 (a being 1000). 



a. 439. The place at which its ordinate is a maximum. 



/3. 468. The centre of gravity. 



y. 500. The centre of the axis. 



S. 577. The point into which the whole contents might be 

 concentrated without altering the attraction on A. 



s. 585. The centre of a sphere of equal solid contents (e A 

 being its radius). 



$. 600. The centre of a sphere with equal attraction on A, 

 a point on its circumference. 



Canonbury, May 11, 1830. . Samuel Sharpe. 



XLIII. An 



