DERIVATION AND ANALYSIS OF METHODS. 261 
nates of a United States Government tug are used in the calculations, and 
the method itself is identical with that used by the French government. 
The signs = and 22 are placed at the side and bottom of the columns so 
the reader can easily follow the operations. I may remark here that the 
trapezoidal method of summation is far more versatile and extensive in its 
application than Simpson’s rules, but depends for its accuracy upon a careful 
correction of the end ordinates. It is taken for granted the reader will 
correct the end ordinates before putting the figures into the formule. It 
is also to be pointed out in particular that in formula (6) the first and last 
terms involve the values 
32, and 42,. 
Therefore note that 
OB eMail los to opp ain ros 
For easy reference the figures (+) are put at the four corners of the 
table, and (3) at the four sides, to indicate that portion of the end ordinates 
to be entered into the columns. 
We may now assemble together the working formule: 
S = 2wDy (a) 
A = 2szy- (b) 
V = 2swrZy (c) 
V 
ie 35 o 
eee (e) 
420 
Tables I, II and III are used to calculate these results and the three 
tables are here given separately to show the method of derivation, and for 
the sake of simplicity of explanation, but may be combined, for operation, 
into a single form. 
Table I gives the 2 functions of water-line areas by summing hori- 
zontally, and the 2 functions of sectional areas by summing vertically; 
the summation of these functions, separately, in the lowest line, or the right 
column, gives a check on the value of the 22 function of the displacement. 
The most noteworthy thing about this table is its simplicity due to the small 
size and number of figures used. : 
The Subtractive Method.—In ‘Table I we have found 22;, which is the 
function of the volume of displacement to the fifth, which is the upper, 
