DERIVATION AND ANALYSIS OF METHODS. 263 
That is, to get from any water-line to the next lower water-line, we 
must subtract the half sums of the upper ordinates in each case. If these 
subtractions be made, it will be the equivalent of removing the slice of the 
underwater body of the ship between the fifth and fourth water-planes; 
and the sum of the results, after these subtractions, will be 22., the function 
of the volume of displacement up to the fourth water-line. If we apply the 
same method to 22,, 22; may be found, and so on to the lowest water-line. 
It is necessary to apply these end subtractions back to the ordinates 
when we have to subtract moments as well as volumes; and this subtractive 
method is subject to be applied throughout all the ship calculations where 
applicable. For the purposes of this paper, however, we will avoid the 
question of moments, and consider merely the calculations for volume of 
displacement. In that case, we do not need to use the ordinates, but may 
find 22, directly from =2;; thus 
pp,a3p,—2et 2s 
= g21 + 22+ 23+ 324 
and so on for the functions of volumes of displacement to the other water-lines. 
TABLE II.—HavINc THE FUNCTIONS OF Vs FROM TABLE I, TO FIND FuNc- 
TIONS OF V1, Vs, V2, Vi. THAT IS, FROM 22s FIND DD4, DLs, DZ2, AND ZDi. 
Subtract 
from 
column 7. 
Subtract Subtract Subtract 
from 
column I. 
Function Function 
of Vo. 
Last line. Function Function 
Table I. 
from from 
Ea) cote guilt ae 
Vee Veolia a 
If we should desire to know 2s, Ds, D3, D2, and D1, which are the functions 
of sectional areas up to the fifth, fourth, third, etc., water-lines, we can 
take the values of Zs from column 7 of Table I, arrange them in column 
similar to Table II, and obtain the functions 2, etc., by subtracting succes- 
sively the half sums of the upper ordinates each time. This is mentioned 
because the areas of sections to each water-line are found in the American 
calculations. In the French method, only the areas of sections to the 
highest, or the load water-line are found, being sufficient for the purpose 
of design. 
