DERIVATION AND ANALYSIS OF METHODS. 273 
If, however, we prepare combination tables in which are included the 
usual ship calculations, we may use any one of the above formule so as to 
fit to the forms most conveniently. 
TABLE VII.—LONGITUDINAL POSITION OF CENTER OF BUOYANCY BY FouR 
MEtHopDs. STEAM TuG OF TABLES I To V. 
Dy from S from 
Biamiber cf Table III, ss Table III, 
4 column 2. column 3. 
() 0.3 
—1108. 
Z(n=zy) 2 EnS=— 396. 
Sz (nZy) _99.0X9.17 sEnS _ —396X9.17 
ZS 984.4 
; Ss = 9.17 feet 
£ 9-17 =i x=—3.69 
V5= 9027 cubic feet. 
from Table III. 2s'w2(nZy) _ —99X(9.17)?X4 Sms _ —396X (9.17)? 
V 9027 V 9027 
x=—3.69 x=—3.69 
The foregoing example finds the value of x for V; only, and if its value 
for a lower water-line is desired, we have but to add or subtract the half 
sums of end ordinates successively in reducing to the next water-line below. 
2. Vertically—We use the symbol z to denote its distance below the 
load water-line. The volume being generated by a water-line area A 
moved in the direction z, an element of volume is Adz; and this multiplied 
by its distance z, gives its moment about the load water-line. The sum 
