*8 6 
Mr. Knight on the Attraction of such Solids 
species, it is required to determine the nature of the curve pr, 
fig. 15, so that the solid may have the greatest possible action 
on a point at its vertex p. 
If we put x for the distance of the generating polygon from 
the vertex, andy for the perpendicular let fall from the centre 
of the polygon on one of its sides, the action of the solid is 
2 nfx arc (tang. = y v/i 4 - -f-y') — / (» — 2) ttx 
by Prop. 5, and the mass of the solid is nrfy*x ; so that the 
quantity whose fluxion, with respect toy, must = 0 , is 
I J _ I 
un arc (tang. = —\/ 1 4 — — y*) 4" an d we get 
4- C = 0 , or x 4 — C 2 * * * (x 9 4- y 1 ) 2 (x 3 4* 
(•* 3 + y z ) VV+ (1+ r^y* 
= »• 
Let a be the value of x when y — 0, then C 4 = and the 
equation of the curve becomes 
a 4 x 2 - (x*4-y 4 ) 2 |x 2 4- (1 4-r }y 2 ] — 0. 
When the polygon is a circle, r — 0, and the equation is re- 
duced to a A x 2 — ( x 4 4 - y 2 ) 3 = 0, the same as we found in Cor. 1 , 
Prop. 31. 
Lemma 1. 
To find the attraction of the right prism whose base is the 
triangle mrv, fig. 1, and height d, on the point p, in the direc- 
tion pm ; on the supposition, that the density at the ordinate 
ks is as any function of the absciss mk, and distance pm. 
If we use the same notation as in Prop. 1, and put f (a, T) 
for the density of a particle any where at the line ks, we shall 
find, by proceeding, as we did there. 
arf (a.T) TT 
