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PIERHEADS, QUAYS, AND LANDING-STAGES.
217
If N be odd, the displacement for the middle compartment is zero.
Accordingly, we have the following series to summate :—
^{(N-l)*+(N-3)*+(N-5)*+.................
which cornes to
AL2 N(N2-1) AL2/. 1\
N3' 6 6V NV
The area of the original rectangle is LD, and, dividing the preceding
expression by it, we obtain the following value for the horizontal component
of transference :—
6D\ NV
Similarly, it can be shown that
△Yt
V 6DV NV
Comparing these equations with the values on p. 205 previously given for
the ordinates in the case of the undivided and unballasted pontoon, we see
that they differ only by the factor
which is a constant for any
assigned case. Therefore, so long as the displacement is confined within
limits such that the bottom of the pontoon is not exposed either inside or
out, nor the upper corners immersed, the curve of buoyancy is parabolic as
before.
The metacentric height, measured above the primary centroid of buoy-
ancy, is
\12D 6D/\ NV’
or, measuring from the centre of depth O of the pontoon,
(^X^M^^)]
When in the upright position, A = 0, and the latter expression reduces to
from which it is clear that with an increase in the number of compartments,
the metacentric height increases, and therefore the greater the number of
the compartments the greater the stability.
When N = 1, the metacentric height is
and as the metacentric height (measured, as assumed above, from the centre