224 Prof. F. Elgar. The Variation of 



tical in cross section. The shorter axis is vertical when the bodies 

 are upright, and is two-thirds of the longer axis, or extreme breadth. 

 The measurements in the direction of OY give the depths of immer- 

 sion, and those in the direction of OX represent moments. The curve 

 APO gives the values, for the rectangular body when inclined to an 

 angle of 30 from the upright, of the horizontal shift of the centre 

 of buoyancy multiplied by the immersed volume, or BB x V, see fig. 8. 

 Thus if Oa be any draught of water, the ordinate ax gives the value 

 of BB x V at that draught. The curve AQO is the corresponding 

 curve of moments for a prismatic body of elliptical section, and of equal 

 length to the above, when inclined also to an angle of 30. The axes 

 of the ellipse are of the same length as the sides of the rectangular 

 section taken in the former case ; the minor axis being two-thirds of 

 the major, and the minor axis being vertical when the body is upright. 

 Apo and A.qo are the corresponding curves of GZ x V at the various 

 draughts of water, and are obtained by deducting VxBGsin30 

 from the ordinates of the carves APO and AQO. The bodies 

 being homogeneous, G is taken at the middle of the depth. These 

 curves therefore represent the actual righting moments of the two 

 bodies under consideration, for an inclination of 30, at all draughts 

 of water. The ordinates measured to the right of AO giving righting 

 moments, and those to the left, if there were any, would be upsetting 

 moments. It will be seen that the whole of the curves in fig. 11 are 

 symmetrical with respect to a line drawn parallel to OX at one half 

 the depth of total immersion. The elliptical prism tends to return to 

 the upright, when at the inclination of 30, at all draughts of water ; 

 and exerts the maximum effort to do so when immersed to the 

 middle of its depth. The rectangular prism, when inclined to the 

 same angle, also tends to return to the upright at all depths of 

 immersion ; the maximum righting moment is not, however, obtained 

 when floating at the middle of its depth, but at draughts which are 

 at equal distances above and below it. 



Fig. 12 represents similar curves for a prismatic body, the upper half 

 of whose section is rectangular, and the lower half elliptical as shown 

 in fig. 6 ; the extreme dimensions of the section being the same as in 

 the previous cases. This form of section is an example of the kind of 

 departure from symmetry of form which exists in ships. It has been 

 seen that if homogeneous bodies of symmetrical form be altered in 

 density so as to float alternately at water-lines which are at equal 

 distances above and below the centres of such bodies, the righting 

 moments at equal angles of inclination will be the same at each 

 draught. In the body for which the curves*in fig. 12 have been con- 

 structed, the departure from similarity of the immersed and out of 

 water volumes causes a difference between the righting moments at 

 the draughts described. APjO and AP,0 represent the curves of 



