1867.] 



the Statical Stability of a Ship, 



333 



of the plane of floatation at the indination. This, divided by the 

 unaltered displacement, gives the radius of curvature required. 



But the chief practical difficulty lay in finding the means of drawing an 

 inclined water-line across the body plan, so as to give an unaltered dis- 

 placement. This I have at length succeeded in overcoming, as follows. 



The sheer- draft calculation gives us, inter alia, the areas of the level 

 sections, belonging to the upright position, as rectangles. Now, if we 

 make one side of each of these equal to the length of the ship, their 

 breadths form a series of ordinates for a curve of mean section ; that is to 

 say, the transverse section of a cylindrical body, of which the displace- 

 ment at any level immersion will be the same as that of the ship. We 

 then make out a scale of displacement for this section at various immer- 

 sions, for a selected inclination, taking care to measure the immersions on 

 the middle line of the original body plan. By this means the finding 

 of any water-line at the selected inclination is reduced to a problem of plane 

 geometry ; and it is obvious that the place of the water-line so found will 

 be a very close approximation to that of the required plane of floatation in 

 the ship. 



The calculations are as follows : — 



1 . Take out the horizontal areas from the sheer-draught calculation, and 

 divide each by the ship's length. Set them ofl^ right and left from a 

 vertical line at their present vertical interval, and draw a curve through 

 their ends. 



2. Any practised draughtsman will have little difficulty in drawing, at 

 sight, an inclined line of floatation which shall give an unaltered immersed 

 area on this mean section. He can verify it by measuring the immersed 

 and emersed triangles obtained by his first guess, and make the correction 

 due to the difference, if they do not agree. 



3. In strictness, the more accurate course would be this, — through each 

 of the vertical stations draw right lines at the selected angle. Thence, by 

 Simpson's rule, form a scale of areas, ending at the highest inclined water- 

 line. Use the vertical interval of the upright displacement, and neglect 

 the cosine of the inclination. Then divide the upright displacement by 

 the ship's length and by the cosine of the inclination, and find to what 

 immersion this displacement corresponds in the scale of inclined areas. 

 But this is needless, unless the calculations have to be made for different 

 draughts of water. 



4. Use this immersion to draw the inclined plane of floatation in the 

 body plan. 



5. Calculate the area, common moment, and moment of inertia of this 

 plane, about the longitudinal axis formed by its intersection with the 

 original plane of floatation, upright. 



6. Transfer this moment of inertia to the longitudinal axis passing 

 through the centre of gravity of the inclined plane of floatation. 



7. Divide the moment so found by the displacement. This will give 



