DERIVATION AND ANALYSIS OF METHODS. 277 
z is the distance of the center of gravity of the outside plating below 
water-line, where water-line usually represents the load water-line. This 
finds the vertical position of the center of gravity. It is well to remember 
we are dealing with surface and not volume, and not confuse the two, as 
is frequently done in some text-books in vogue. 
To find the longitudinal center of gravity of outside plating, however, 
we have the elementary moment ydx.x and derive the value of « by the same 
method as in the case of a curvilinear area, previously explained, viz., 
sany 
= (2) 
x = 
keeping in mind the difference in the nature of the ordinates. 
Some authors correct the half-girth ordinates, increasing them slightly 
to compensate for the additional surface not developed in expanding the 
plating transversely, but not longitudinally. This may be done but the 
result will be sufficiently accurate without it, particularly with regard to 
the position of the center of gravity. ‘The idea in correcting ordinates is 
to be able to get the true surface at the same time, by which to get the weight 
of the plating. It is much more accurate and expeditious to calculate the 
center of gravity from the simple, uncorrected half-girths, which makes 
an inappreciable error in the location of center of gravity; and to calculate 
the area of the surface from a more exact method, since the error in weight 
by the former method might be more appreciable. 
Example.—A ship is divided into fifteen sections at a common interval 
of 16 feet apart. The half-girths are as entered in/Table IX. ‘The center 
of gravity of each half-girth is found by the foregoing graphic method, 
and its distance z, below the load water-line is entered in the table. 
Find the vertical and longitudinal positions of the center of gravity of 
bottom plating. 
The figures of Table IX may be further simplified by using an 
auxiliary water-line below the load water-line, but just above the 
highest position of any of the values of z. Suppose the fifth water-line falls 
at 7 feet below the load water-line, we can locate z with regard to this 
water-line by measuring the various values of z below it. That is the 
equivalent of subtracting 7 feet from each value of z. The values of z to 
be used in the table will be 0.3, 1.7, 2.9, 4, 4.9, 5-2) 5-7, 5-7) 5-5) 5-1, 4.6, 3.9, 
3.5, 3-0, instead of the larger figures. This simplifies the multiplications 
and additions in the column yz, by handling smaller figures. ‘To the vertical 
distance thus found we can add 7 feet to get the distance from the load 
water-line, which will give the same result as in the table. 
