Mr. Sharpe on the Solid of greatest Attraction. 1 1S1 



cl- ra cz, and c : a : : a : z and z : c : : c : x; from the first pro- 

 portion z may be drawn, and from the second x. 



On the diameter A B (fig. 2.) draw a semicircle A C B, and 

 from A draw the 



chord A C pro- Fig. 2. E 



duced till it meet 

 a perpendicular 

 from B at D; then 

 if A C be = c, 

 A D = z. On 

 the diameter pro- 

 duced make A E 

 = A D ; on A E 

 describe a semi- 

 circle APE, and 

 with the centre A 

 and radius A C 

 describe a circle 

 cutting A P E in 

 P; P will be a 

 point in the curve required : for letting fall P M on A B, 



A P or A C : A B : : A B : A D, and 

 ADor AE: AP:: AP:AM; 



and if several other points P be thus laid down, the curve 

 will be drawn. Q. E. D. 



2. The solid is obviously made by the revolution of the 

 area APB on the axis AB: now to measure that area, 



'. ',' , 4 4)1 1 



y x the fluxion of that area = a? — x^ 2 x 3 x whose 



fluent corrected = 



4 



a? - 



4 3 



r which when x = a be- 



comes -£- = the area A P B M. 



3. To determine the solid contents; let p = 3*14159 &c. 

 the area of a circle whose radius is unity; then py*x is the 



4 5 



fluxion of the solid whose fluent is f p a 3 ' x^ — ^ p x 3 which 

 when x becomes a is = —~ = the whole solid contents. 



15 



4. Now since the contents of a sphere with radius b is=^-> 



by making ~— = 



4 pa 3 

 15 



we obtain a = b 



855 



or b = a 



6 

 1170 



6 15 ~ " 1000 " 1000' 



being the proportion between the axis of this solid and a 

 sphere of equal contents. 



N.S. Vol. 8. No. 46. Oct. 1830. 2 L 5. The 



