re 
a 
DERIVATION AND ANALYSIS OF METHODS. 279 
The wetted surface may be calculated by this formula, using the proper 
value of y, which is generally accurate to within two tenths of one per cent. 
MOMENT OF INERTIA AND METACENTER. 
Let us take the usual nomenclature, 
I = moment of inertia about a given axis. 
I, = moment of inertia about a parallel axis through center of gravity 
of the figure. 
h = depth of the figure in a line perpendicular to the axis. 
d = perpendicular distance between the axes corresponding to J and Jo. 
A = area of the figure. 
We know from the text-books the following equivalents 
h == Al (1) 
and 
I=1,+ Ad’. (2) 
1. Longitudinally.—If we take the moments about the y axis, of the 
elementary areas ydx, we will have 
fe fe ydx (3) 
The term corresponding to the value of Jo drops out of (3) because the 
area of the strip is so small, and the value of /?, which is equal to dx’, is so 
small that Jo is negligible. 
If now, for calculating purposes, we divide the area into a number of 
sections by ordinates having a common interval, s, the working formula 
will become 
I = sXx’y (4) 
x is the distance of the ordinate from the y axis, and if we let 7 be the 
number of intervals the ordinate is removed from this axis, we will have for 
any given ordinate x = us; that is, for ordinates 1, 2, 3, 4, etc., stations 
away from this axis, the value of x is s, 25, 3s, 4s, etc., respectively. Thus 
x? = n’s?, in which s is constant; therefore, the formula becomes 
Ls ny (5) 
