SIMPSON'S RULES 353

only three ordinates the multipliers would be 1—i—1; if we had five
ordinates 1—£—2—i—1; and if more than five 1— 1—2—i—2—i—1.

The sum of these products multiplied by one-third of the common
interval gives the area enclosed by the curved line A C B and, as this is
half the vessel's waterplane, by doubling it we obtain the area of the
whole waterplane.

An easy way to remember the multipliers is to write the figures 1,4,1
in groups of three against the ordinates as follows:—

1-4—1 1—4— 1

add 1-

and these are the multipliers for the respective ordinates.
Example —Find the area of the waterplane of a barge, given length
of waterplane=m feet. Ordinates in feet=3,10,16,19 5,21,19,15-5,
10, 6. As we have nine ordinates given, the common interval is one
eighth of the length=15-5 feet, thus in Fig. 1 the distance between a
and b and between b and c=15-5 feet. The calculation of the area
A C B is then carried out as follows:—
Ordinate Length of Ord. Simpson's Multipliers Products
a 3-0 1 3*0
6 10-0 4 40-0
c 16-0 2 32-0
d 19-5 4 78-0
e 21-0 2 42-0
/ 19-0 4 76'0
g 15-5 2 31-0
h 10-0 4 40-0
k 6O 1 6-0
348-0
348X15-5
Area of A G B =----------- = 1798 square feet
o
Area of waterplane=l798x2= 3596 square feet
Simpson's Second Rule.—Is applied when the area is divided into a
specific number of equidistant ordinates which must be 4 or 7 or 10 or