Page 84 - statbility for masters and mates

72 Ship Stability for Masters and Mates
Then area Aa0 B a0 a1h+a2h2 C a0 2a1h 4a2h2
a0 A B C a1h B 2C a2h2 B 4C
Equating coef®cients:
A B C h, B 2C h=2, B 4C h=3
From which:
A 5h; B 8h; andC h 12 12 12
;Areaoffigurebetweeny andy 5hy 8hy 1 hy 1 2 12 1 12 2 12 3
or
This is the Five/eight (or Five/eight minus one) rule, and is used to ®nd the area
between two consecutive ordinates when three consecutive ordinates are known. Summary:
A coef®cient of 1 with multipliers of 5, 8, 1 etc. 12
Areas of water-planes and similar ®gures using extensions of Simpson's Rules
Since a ship is uniformly built about the centre line it is only necessary to calculate the area of half the water-plane and then double the area found to obtain the area of the whole water-plane.
Fig. 10.4
Figure 10.4 represents the starboard side of a ship's water-plane area. To ®nd the area, the centre line is divided into a number of equal lengths each `h' m. long. The length `h' is called the common interval. The half-breadths, a, b, c, d, etc., are then measured and each of these is called a half-ordinate.
Using Simpson's First Rule
This rule can be used to ®nd areas when there are an odd number of
Area h 5y1 8y2 y3 12
ordinates.
Area of Figure 10.5(a) h3 a 4b c