294 
Mr. Knight on the Attraction of such Solids 
Another Method of finding the Action of the Tnangle vrm. 
The expressions we have found will terminate only when 
m is one of the series of numbers 2, 4, 6, &c. If m is among 
the odd numbers 1, 3, 5, &c. will be a whole positive 
number ; and we have for the action of the triangle, or rect- 
angle (accordingly as the fluent, with respect to t, is taken 
from t = 0, to t = rT, or from t = 0, to t = y ' ) provided m 
is greater than 3, 
a _____ r ^ nr\ J I _ ^ f Wl — 2 
J aI X(m — 1) (a z + T 2 ) X , a * t 1 (to— i) (m— 3) (a z + T 2 ) z 
2 
r 
n — Q B 
(to— 2) (j« — 4) 
' (m-i) (m-3) (m- 5 ) (^-j T 1 j t*f ~ 5 
4" 
4* 
(m- 2) (m — 4) 3 
(ni — 1) (to — 3) (m — 5) 2 («’-■-}- 7’ 2 )- — i a z ~[-T--> r t 
+ 
(m — z) (to— 4) 3 
(to— 1) (m— 3) (TO-5) 2 (« a -f T 1 )' 
nz — 1 
2 
X 
Va^-iT 
== x arc (tang. = 
1 \ O 
£ 
vV-j-T 
Let us, for brevity, denote this quantity b yfaf® (a, m, T, t ) ; 
then for the action of the triangle vrm we have A = faT<p 
(a, m, T, rT); and for the rectangle, whose sides are rm (y) 
and rv (y 1 ) A = fa fcp («, m, T,y'). When m = 1, or 3, the 
above expression will not give the attraction ; but we evidently 
have, in the case of m = 1, 
A = f x arc (tang. — ■ — ? * ) ; and when m = 3, 
v/ vV + T a ^ & 09 
A — /^{ 2 ( a q.p) x rt * + :r a +i 2 ) “I" 2 (a 2 -j- t 2 )! x aic ( ian S — 
