27 2 
Mr. Knight on the Attraction of such Solids 
In the latter case n is infinitely great and r infinitely small j 
r is also the tangent of ■— = ~ ~ . if we reject the 
other terms of the expansion, on account of their smallness. 
The expression for A may then, in this case, be put into the 
form 
. 7 , 2kvr ink 
A = »*”"+ -TV— — ~ T’ 
‘ IT r ~ ~ \ 
nowr{£ + j-v}=( < I-P) r {v + J-V r ‘} = r ( 1 + f)v 
whence the second term of the last number is changed into 
2 hi 
f r% 
ikn ( 1 — “ 
2 hi 2 t 2 nk 2 7 , 
• — knr = — h r, by 
r 3 r 3 J 
putting 7T for wr: and, by substituting this value, we have at 
last A= -j kn; which is the well known expression for the 
attraction of a sphere on a point at its surface. 
If the generating polygon is a square instead of a circle, 
r = tang. 45°= 1, and equation (a) gives A ==: 4# (7r — 2) 
= jfz x 1,14159, &c. which exceeds the attraction of the sphere 
by about one-tenth, if pr is the same circle in both. 
Cor. If we would know the radius {k') of a sphere, which 
shall attract, a point at its surface, as much as a polygonal 
sphere, of the length 2k, does a point at its vertex, we have 
only to put 
— irk' = 2 irk — whence ' 
