DERIVATION AND ANALYSIS OF METHODS. 265 
Likewise, having found 
Ne 
we have 
22_= 22+ 2% 
and so on. 
Tables IV and V give the additive method of calculating the same 
quantities as previously found by the subtractive method, using the same 
ordinates and obtaining the same results. (See pp. 266 and 267). 
The latter method has been explained first because of its simplicity, 
and it leads to a quicker understanding of the additive method. 
In Table IV, the numbers of sections are placed in the left hand 
column, and the numbers of water-lines in the upper line as before, except 
that the lowest water-line is placed first, on the left side, since we work 
from the first water-line, upward, and not from the fifth downward as in 
Table I; so the order of the water-lines is reversed in the two methods. 
We also include the additional columns, with even numbers, 4, 6, 8, 
etc., in which to sum the ordinates as we proceed. 
In the subtractive method it is easy to subtract the half sums of top 
ordinates, but in the additive method it is more complicated, and the half 
sums would have to be worked out separately on a different piece of paper. 
To obviate this, twice the trapezoidal sums of ordinates is found by adding 
successively the sum of the upper two ordinates. Thus 
224 = Vit 292+ 293+ Vs, 
225 = 224+ (os oF Ys). 
This facilitates the additions of ordinates, and the double function 
2> can be used directly in the formula S = w(2Z,). 
The same method is employed to get volumes, for similar reasons, and 
instead of finding 221, 222, etc., as before, we find 2221, 2222, or double the 
functions of volumes, which can be used directly in the formula V = sw(2zZ). 
For this reason it is not necessary to enter the one-fourth and one-half 
ordinates, as in Table I, but we must enter the whole ordinates in each 
odd-numbered column. From the sums of these columns we subtract one- 
half the sums of end ordinates to get Z:, 22, etc., for the water-lines. 
From these values we build up the values of 222 thus 
2 — 
222, = 2,+ 2, 
2223 = 2, + 22. +2; 
2224 = 2, + 22, + 223 + XR, etc. 
