Page 38 - Introduction to Naval Architecture
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26 SHIP FORM CALCULATIONS
Where there are large numbers of ordinates the arithmetic in the table
can be simplified by halving each Simpson multiplier and then
doubling the final summations so that:
Other rules can be deduced for figures defined by unevenly spaced
ordinates or by different numbers of evenly spaced ordinates. The rule
for four evenly spaced ordinates becomes:
This is known as Simpson's Second Rule. It can be extended to cover 7, 10,
13, etc., ordinates, becoming:
A special case is where the area between two ordinates is required when
three are known. If, for instance, the area between ordinates y 0 and ji
of Figure 3.3 is needed:
This is called Simpson's 5, 8 minus 1 Rule and it will be noted that if it is
applied to both halves of the curve then the total area becomes:
as would be expected.
Unlike others of Simpson's rules the 5, 8, -1 cannot be applied to
moments. A corresponding rule for moments, derived in the same way
as those for areas, is known as Simpson's 3, 10 minus 1 /Iwfeand gives the
moment of the area bounded by $> and yi about yo, as:
