the Stability of Ships. 265 
a parabolic curve line is supposed to be drawn through the 
points A, B,C, D, of the 3d dimension, such as the arc of a cubic 
parabola, the ordinates of which are parallel to the axis of the 
curve, and the area terminated by this curve line is assumed 
to approximate to the given area A A' D'DA : by this rule, the 
area = A -fi 3 B x in which expression A =■ a -}- d> B = b -j- c, 
and R — 3 r, which being substituted for their respective values, 
the area AA' D'DA = a + 3 b + S c + d x — . 
In order to bring these rules into a form convenient for prac- 
tical use, let it be proposed to measure the area AA'G'GA (fig* 
24.) intercepted between the extremes of 7 ordinates. 
1st. Suppose the right lines AB, BC, CD, &c. to be assumed, 
instead of the curve lines AB, BC, CD, as terminations of 
the space to be measured : then the area AA'G'GA will be 
equal to the sum of six trapeziums ; AA'B'B, BB'C'C, CC'D'D 
and so on. 
The area of the trapezium A A' B'B = a -|- b x ~ , by the 
rule opposite 2 ordinates : by the same rule, the area of the 
trapezium BB'C'C = b -f- c x ^ ; the area of the trapezium 
CC'D'D = c+dx j, and soon. By adding these six sepa- 
rate areas, the sum will be the area of the space AA'G'GA 
~ a 2h-\-Qc-\-2d-\-2e-{-2j-\-gxh-. The law of con- 
tinuation for a greater number of ordinates is obvious. This 
rule is precisely the same with that which is given by M. 
Bouguer, in his work entitled “ Traite du Naviref * under 
a form somewhat different : his rule is this ; from the sum of 
all the ordinates subtract -§• of the sum of the first and last ; the 
* Page 1 1 2. 
MDccxcvm. M m 
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