Of GREATEST ATTRACTION. 201 



IX. 



1. If it be required to find the equation to the fuperficies of 

 the folid of greater! attraction, and alio to the fections of it pa- 

 rallel to any plane parting through the axis ; this can readily be 

 done by help of what has been demonftrated above. 



Let AHB (Fig. 4.) be a feclion of the folid, by a plane through 

 AB its axis. Let G be any point in the fuperficies of the folid, 

 GF a perpendicular from G on the plane AHB, and FE a per- 

 pendicular from F on the axis. Let AE zz x, EF ■= z, FG — v, 

 then x, z, and v are the three co-ordinates by which the fuper- 

 ficies is to be defined. Let AB =z a, EH — y, then, from the na- 



ture of the curve AHB, y 2 — a 3 x 3 — x 2 . But becaufe the plane 

 GEH is at right angles to AB, G and H are in the circumfer- 

 ence of a circle of which E is the centre ; fo that GE = EH 



=/. Therefore EF 2 + FG 2 ~ EH 2 , that is, z 2 -f v 2 =/, and by 



fubftitution for y* in the former equation, z 2 -f- v 2 = a 3 x 3 — x 2 , 



or (x 2 -f- z 2 + vj =: a* x*, which is the equation to the fuperfi- 

 cies of the folid of greateft attraction. 



2. If we fuppofe EF, that is z, to be given — b, and the fo- 

 lid to be cut by a plane through FG and CD, (CD being paral- 

 lel to AB), making on the furface of the folid the fedtion DGC ; 

 and if AK be drawn at right angles to AB, meeting DC in K, 

 then we have, by writing b for z in either the preceding equa- 



tions, b 2 -f- v* = a 3 x 3 — x z , and v 2 — a 3 x 3 — x 2 — b x for the 



equation of the curve DGC, the co-ordinates being GF and 

 FK, becaufe FK is equal to AE or x. 



This 



