Of GREATEST ATTRACTION. 201 
IX. 
1. Ir it be required to find the equation to the fuperficies of 
the folid of greateft attraction, and alfo to the fections of it pa- 
rallel to any plane pafling through the axis ; this can readily be 
done by help of what has been demonftrated above. 
Ler AHB (Fig. 4.) be a fection 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 =x, EF= 2, FG = VY, 
then x, z, and v are the three co-ordinates by which the fuper- 
ficies is to be defined. Let AB = a, EH =y, then, from the na- 
ture of the curve AHB, y*= a? #3 —x*s) Bat 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 thatGE = EH 
=y. Therefore EF*-+ FG* = EH’, that is, x°--+- v= ‘ and by 
fubititution for y* in the former equation, z*-+ v? = =a? xi — x’, 
or (3° * tu y= = a+ x’, which is the equation to ee fuperfi- 
cies of the folid of greateft attraction. 
2. Ir we fuppofe EF, that is x, to be given =4, 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 fection DGC ; 
and if AK be drawn at right angles to AB, meeting DC in K, 
then we have, by writing 4 for x in either the preceding equa- 
~ 
2: 4: 2 
tions, 4° + v3 = 4? x? —x’, and v'=a? x? —x*—Zd* for the 
equation of the curve DGC, the co-ordinates being GF and 
FK, becaufe FK is equal to AE or x. 
THIs 
