260 SHIP CALCULATIONS; 
where dz and dx are the vertical and horizontal intervals, respectively. 
Integrating the above we have 
V=w2A =s25S. (2) 
In (2) 2 is the sign of summation either by the trapezoidal rule, or 
Simpson’s rule, or any other rule. The trapezoidal rule is used by the 
French and United States governments, and Simpson’s rule by the British. 
In practice V is not usually determined from the equations in (2) but 
directly from the ordinates. To decompose (1) into terms of its ordinates, 
A= 2 f ydx 
and 
S=2 f ydz 
where y differs in the two, being summed in the first instance along the 
water-line area, and in the second, down the sectional area. These equivalents 
thus give us a valuable check on each other in practice. Substituting either 
the value of A or of S in (1) we get the same value for V, thus 
V=2 aa ydxdz. (3) 
If now we integrate, by summation, for stations dx, a fixed distance, 
s apart; and for dz a distance w apart, we have the practical form, since s 
and w are constants 
V =2 sw2dy. (4) 
Although = is the sign of summation by Simpson’s rule, or other rule, 
the author prefers the trapezoidal method of summing areas, and in this 
paper it will be taken to represent the trapezoidal sum, which means that 
26) Ss BaP SEI oo 6 2 Ment Pio (5) 
and 
ZZy = FZ + V2t Uys... . Lar tglyn (6) 
That is to say 2Z represents the trapezoidal sum of the trapezoidal sum of 
the ordinates, and as derived in (3) may be summed either way, by sections 
or water-lines. 
Table I gives a practical example showing the method of calculating 
the value 2Zy preliminary to determining the value of V in (4); the ordi- 
