and 
Mr. Atwood’s Disquisition on 
4 * 3 
^ a c 
+ 
2 56 
therefore, rN = p t 3 a — P * sec - s 
sin. S 2 x cos. 1 S 1 5 
Lr = rN — LN = 
+ 3 " 
2 ' S * 
3 a P X tang. s S . 
and, since LE 
rE = 
p x sec. 2, S 
3 a 
5 ’ 
X tang. 2, S 
4 * 
or rE = -i- X sec . 2 S + 1, and, since the angle 
4 
ErT — S, ET = ■£■ X sec.* S + i, 
or ET = p * tang - - - x cos. S - 4 - sec. S. 
4 
This is the value of the line ET, when the area represent- 
ing the volume immersed is terminated throughout by the 
parabolic arc, the said area being = or -j- x BA x DL ; 
but, if that form should extend to the sides AH, BC, only, the 
remaining part of the volume immersed being of any other 
figure,* and this entire volume should be of any magnitude 
V, the value of ET corresponding will be p * x-~ 
X cos. S + sec: S, _ or ET = x cos. S + sec. S. And, 
since ER — d x sin. S, TR, or the measure of the ves- 
sel’s stability, GZ = x cos. S -j- sec. S — d x sin. S ; 
precisely the same quantity which measures the stability at 
the angle of inclination S,-f- when the sides are parallel to the 
masts above and beneath the water-line : a coincidence not a 
little remarkable, and such as would not probably have been 
supposed to exist, except from the evidence of demonstration. 
From this proposition it is inferred, that if the sides of a 
f Case 1. page 216. 
• See pages 213, 214 . 
