270 
Mr. Atwood's Disquisition on 
RULE III. 
Fluent of Zz = S -f 2 P + 3 O x 
in which expression, 
S = the sum of the first and last ordinate. 
P = the sum of the 4th, 7th, 10th, 13th, &c. ordinate, (the last 
excepted. ) 
Q = the sum of the 2d, 3d, 5th, 6th, 8th, 9th, &c. ordinate. 
r = the common distance between the ordinates. 
It is to be observed, that the first of these rules approximates 
to the fluent, whatever be the number of given ordinates. The 
second rule only requires that the number of ordinates shall be 
odd. To apply the third rule, it is necessary that the number 
of ordinates given shall be some number in the progression 4, 
7» 10 > 1 3 ’ & c - the number of ordinates must be a mul- 
tiple of 3 increased by unity. But, in every case, the approxi- 
mate fluent may be obtained, either from the Rule 11. or the 
Rule hi. or by employing both rules conjointly. 
Before these theorems are applied to practical mensurations 
in naval architecture, it may be satisfactory to examine, by a 
few trials, to what degree of exactness they approximate to the 
correct values of curvilinear spaces. This will be known, if the 
area of some curve, which is exactly quadrable by other geome- 
trical rules, be measured by them. Such as a parabolic figure of 
which the equation is y s = p 7 x, x being the abscissa coincident 
with the axis, and y the corresponding ordinate perpendicular 
to it. 
The semi-area of this parabola (fig. 25, 2 6.) is known to 
be xy x — ; and the curve is termed a parabola of 8 dimen- 
sions. 
