268 SHIP CALCULATIONS; 
time it requires. In short the subtractive method is much simpler, is less 
liable to error, and requires much less time to work out. ‘The additive 
method, however, combines more calculations on one sheet of paper. The 
subtractive method has another great advantage in brevity. By it we find, 
directly, the various functions to the upper water-lines, which may be 
extended to include the determination not only of the displacement, but 
also of metacenters, moments of stability, changes of trim, and in fact the 
whole ground of ship calculations. This upper water-line is the extreme 
upper limit to which calculations are carried, and in a ship of size and 
importance, from twelve to fourteen water-lines are used, the tenth being 
always the load water-line. It is only in a few smaller vessels, like the tug 
calculated in Tables I to V, that a fewer number of water-lines than ten are 
used. Take, for instance, a ship divided into twelve water-lines for the 
calculations; the values of displacement and stability functions are found 
for the twelfth water-line, at once, as in Table I. Now, by the subtractive 
method, the functions for the eleventh, tenth, ninth, eighth, seventh and 
sixth water-lines are found but no more, since the vessel will never float 
at a draft less than that of the sixth water-line, nor even as small; and so 
it is considered a loss of time to find these functions for the lower water- 
lines. In the additive method, we commence at the first water-line, and 
include every one up to the twelfth, which not only requires the additional 
time and labor, but if an error is made in one of the lower water-lines, it 
is carried on to the upper water-line, and is more difficult to discover and cor- 
rect. Preference is to be given to the subtractive method, for by it the sim- 
plicity and beauties of the trapezoidal rule are utilized to full advantage. 
Displacement Calculations by Simpson’s Rules.—The first rule is almost 
exclusively used, and its application to the formule is precisely the same 
as in the trapezoidal rule except that the signs 2, and 22, have a different 
meaning. 
If Simpson’s rule be used 
Ly = 31+ 492+ 29s + 49, etc.) 
and 
Dy, = 4(21+42.+ 223+ 42,, etc.) 
We can form tables dealing with the ordinates, by this rule, but we 
must include the multipliers, 1, 4, 2, 4, etc., for both ordinates and 2’s, 
which complicates the calculations and increases the size of the figures 
handled. With this modification, Simpson’s rule can be applied to all the 
foregoing formule, where 2 and 22 are used; and in fact these formule 
