Page 35 - Introduction to Naval Architecture

SHIP FORM CALCULATIONS 23

These equations give:

This is Simpson's First Rule or 3 Ordinate Rule.
This rule can be generalized to any figure defined by an odd number
of evenly spaced ordinates, by applying the First Rule to ordinates 0 to
2, 2 to 4, 4 to 6 and so on, and then summing the resulting answers.
This provides the rule for n + 1 ordinates:

For many ship forms it is adequate to divide the length into ten equal
parts using eleven ordinates. When the ends have significant curvature
greater accuracy can be obtained by introducing intermediate ordi-
nates in those areas, as shown in Figure 3.4. The figure gives the

Figure 3,4

Simpson multipliers to be used for each consecutive area defined by
three ordinates. The total area is given by:

where y 1, y 3, y 11 and y 13 are the extra ordinates.
The method outlined above for calculating areas can be applied to
evaluating any integral. Thus it can be applied to the first and second