Stability with Draught of Water in Ships. 221 



the point under consideration, is the following: If any homo- 

 geneous body, which is symmetrical about the three principal axes 

 at its centre of gravity, be of such density as to float with its lowest 

 point at a depth x below the water ; then, if the density be altered 

 so as to make it float with its highest point at a height x above the 

 water, the righting moments will be the same in both cases at equal 

 angles of inclination, and consequently the range of stability and 

 complete curve of righting moments will be the same. This pro- 

 position can be made still more general, as was shown by Mr. William 

 John, in a letter to the " Times " of the 5th September; as it applies 

 to all homogeneous floating bodies of irregular form revolving 

 about a horizontal axis fixed only in direction. In this general 

 form, the condition of turning the body through an angle of 180, or 

 upside down, must, however, be included, because the immersed 

 volume in the one case must be of the same form as the emersed 

 volume in the other, and this can only be obtained with irregularly- 

 shaped figures by turning them through an angle of 180. For this 

 reason I chose symmetrical figures for the purpose of giving a 

 popular illustration of the analogous effects of low freeboard and 

 light draught upon the stability of ships, and avoided introducing 

 the condition of turning through 180 in order to get similar volumes 

 above and below water in each case. The general proof of the 

 proposition laid down is that the Hue joining the centre of gravity of 

 the immersed volume with that of the volume above water must pass 

 through the centre of gravity of the whole body, and the distances 

 of the centres of gravity of the two sections from that of the whole 

 body are inversely proportional to their volumes ; so that the moment 

 of stability, which is proportional to the immersed volume multiplied 

 by the distance of its centre of gravity from that of the whole body, 

 will be the same in both cases. 



As the righting moments at equal angles of inclination at the deep 

 and light draughts described are the same in a homogeneous floating 

 body of symmetrical form, it follows that at the same draughts the 

 lengths of GZ, or the arms of the righting lever at equal angles of 

 inclination, and also the metacentric heights, are in the inverse ratio 

 of the displaced volumes. 



The moment of stability Vx GZ is by Atwood's formula, see fig. 8, 

 equal to v x h-Ji 2 V x BG sin 0. Now v X hji z , which is the moment 

 of the wedges of immersion and emersion, is the same whether 

 WAL be the below- water or above-water volume ; the immersed 

 wedge in the one case being the emersed in the other, and vice versa. 

 V X BR is therefore the same in floating bodies of any form that 

 revolve about a horizontal axis fixed in direction, whether WAL be 

 the above- water or below- water volume. For such bodies as are 

 homogeneous VxBGsintf is also equal in the two cases, because 



