274 SHIP CALCULATIONS; 
of the moments of all the elementary volumes, divided by the whole volume 
will give z. ‘Thus 
M_S Adzs _winA _2sw'Z(nZy) _ 2sw'd(ndy) _ wz(nzy) (1) 
V (Ads wah  2wldy V Sy 
S= 
all of which are derived precisely as for x in the foregoing problem; and, 
as in that case, we have several expressions of this value, either of which can 
be used as best suits the problem in hand. Thus 
Z= wend from water-line areas. (2) 
aS A 
2: 
z= w A from water-line areas and volume. (3) 
Z= ae from ordinates and volume. (4) 
or 
Z= ee directly from the ordinates. (5) 
The operator should make himself thoroughly familiar with these 
formule; they are very simple, and when once fixed in the mind, the ship 
calculations become a very simple matter, and will obviate years in studying 
the rules and tables of various authors, with the accompanying vague 
notion of the reason why the different operations are performed. With 
these formule, any table of ship calculations can be analyzed, whether by 
trapezoidal or Simpson’s rules, by supplying the symbols opposite the figures ; 
and with them any tables may be departed from, and new tables constructed, 
or the calculations can be made, without tables, directly from the formule. 
Probably the greatest loss of time on the part of the operator is in trying 
to work out his problem by following the operations in some similar table 
already worked out, or in a text-book, without always understanding the 
work, which is a source of error and delay. 
Table VIII is subject to the same remarks as for the preceding table. 
The values of 2y must first be found as in Table I. 
- Also, in combination tables, for the next higher or lower water-line, 
the method of adding or subtracting the half sums of end ordinates may be 
employed. 
