270 
Mr Knight on the Attraction of such Solids 
have their sections in one direction continuous curves, whilst, 
being cut in a different way, there results, from their inter- 
section with a plane, a polygon, or rectangle, or some other 
right lined figure. 
As no one, that I know of, has considered the actions of 
such bodies, I shall offer no apology for giving a few ex- 
amples. 
Let uvvV, fig. 15, represent any regular polygon, whose 
plane is perpendicular to the line pm, and its centre in that 
line : moreover, let this polygon be variable in magnitude, and 
move parallel to itself in the direction pm, in such a manner, 
that the middle point r of each of its sides uv, may describe a 
given curve pr* 
Prop. 24. 
Let it be required to find the attraction of the solid thus 
generated by the polygon, when the curve pr is a circle,* and 
the attracted point at the vertex p of the solid. 
The attraction of a regular polygon was found in Prop. 5 
and it will be adapted to our present purpose, by putting ,r e 
for a% and zkx — x* for 6 s , where k is the radius of the circle 
pr: and we have, for the attraction of the solid, 
A = znfx arc (tang. = ~ iff- ~^-[2kx—x 3 )) — (n— 2) ttx 
or A = 2 njx arc (tang. — d\/ x—dx 2 ) — [n — 2) ttx. 
That part of A, under the sign of integration, equals 
2 nx arc (tang. = ^ %/ x — r 2 x 2 ) — znfx arc (tang, 
= — V 2/'? ( 1 -j- / * ) X — l 2 X 2 ) , 
* We may, not improperly, term this solid a polygonal sphere. 
