as are terminated by Planes , &c. 293 
The farther integration, with respect to T, is not necessary for 
our purpose. 
Cor. 1. If we multiply this by 2 n, it will be the attraction 
of a regular polygon of n sides ; and making n infinitely great 
and r infinitely small, the attraction of a circle to the radius T 
is found to be 
1 znraTT 
A =/ 
— znra 
+ (m — 1 ) 
r, + 
znra 
, \ « 
(?7i— i) a 
which, by putting tc for nr, is 
A = 2 
A 
t ( m — i)a L 2 (to — i) (a * 2 -f Z 11 /— 
the same as is found differently by other writers. 
Cor. 2. When r becomes infinite, the triangle vrm is changed 
into a parallelogram, infinitely extended in the direction rv ; 
and we have 
A =/ 
aT 
(a 2 + T9 (rTf 
: { »— ( 2 ~ m ) + (2—®) U-«) 
— &C. 
3 . 5 ta'+'ny 
which, when m is an even positive whole number greater than 
2 . 4.6 (to — 2 ) f* aT 
2, is reduced to A 
3-5*7 1'n—ijiS + 
2 
Cor. 3. If instead of the action of the triangle vrm, that of 
a rectangle, whose sides are rm (y) and rv (y') 5 had been 
required, we must have proceeded exactly in the same man- 
ner, but the fluent, with respect to t, must have been taken 
from t = 0, to t = y'; so that we have only to substitute y* 
for rT , in the value found by the lemma, and there results 
A = /■ 
ayT 
(a*+T*) (a* + T*+y''f- 
(2 —m) + (*-») 
(4 -®) 
3.5 (»»+/•*)* 
&C. | 
MDCCCXII. 
