34 NAVAL SCIENCES. 



under and over the water form receding angles, and in the other salient angles, 

 both being equally acted on by wind and sails, one ship will have the greatest 

 security and the other be exposed to the greatest danger, although the formula 

 for stability gives the same value in both cases. It hence appears that this 

 formula must be used with great caution and judgment. The actual stability 

 must be determined from the given formula, since in most cases the two 

 bodies E and I are not actually equal, and their line of intersection would lie 

 to the wind side of the water-line. Hence an eccentricity of from y\ to j^ 

 of a foot has been assumed in the transverse section of the ship for the line 

 of intersection of these two surfaces. We must, therefore, calculate the 

 contents of the two bodies, whose transverse section is a mixed triangle, one 

 side of which may be regarded without error as a part of a parabola. 

 Having completed this calculation, we must calculate the true contents of 

 the parts immersed and emerged by the inclination, according to the 

 proper formulas, and if it should appear that they are unequal, we must 

 take another point until we obtain this equality. Supposing that we have 

 at length obtained the position of the true inclined water-line, we can pro- 

 ceed to calculate the stabiUty by the (ormulafW Zdx -{-/ wzdx —Dis. 

 The integral of the function WZdx is obtained by the above mentioned 

 sectors ; the different values of Z and z are obtained by calculation, and 

 the values of W and w are found by the following method. Let SBD 

 {pi, 7, fig. 8) be one of the sectors, SD the straight, and SB the inclined 

 water-line. The line DB divides the sector into a triangle and the adja- 

 cent parabolic surface. Bisect BD at E, draw EG perpendicular to BS, 

 and take EF = | of this line. From E and F drop the perpendiculars 

 EG and FH on SB, and | SG will be the distance of the centre of gravity 

 of the triangle SDB from the point S, measured on the surface of the 

 water, and SH the distance of the same point to the centre of gravity of the 

 curved surface DCB. Hence the formula | SG . SBD . SH . BCD gives 

 the value of WZ for this sector, and applying the formula for the equi- 

 distant ordinates (see § 1, p. 31), we determine the integral off WZdx. We 

 make use of the same process to obtain the integral offwzdx. As regards 

 the function Dis, the displacement D has been already calculated, and s the 

 assumed angle of inclination and the element a, which depends on the true 

 position of the centre of gravity, can only be found by calculation or experi- 

 ment with a ship of precisely similar construction. We can hence determine 

 the true measure of the stability by the {oYmu\aL'D.GY=f WZdx -^-/zwdx— Dis. 

 A simple method of finding the centre of gravity of the ship's body has been 

 given by Abethell, who takes his data from docking the vessel, which of 

 course is done at high water, the water passing off with the ebb tide, and then 

 the dock-gates are closed. He takes the time when the extremity of the 

 keel touches the foundation of the dock, as the water passes off. From that 

 time the water gradually leaves the after part of the ship, while the bows 

 are immersed to a greater depth, and an equilibrium takes place between 

 the total weight of the ship and the pressure of the water upon the immersed 

 portion, until the moment when the ship is supported at both ends. During 

 this time the ship is to be regarded as a lever of the second kind, the fulcrum 

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