and their Action on Floating Bodies. 89 



describe approximately the ellipse h e f g. The diameter 

 of the orbital circle is here the major axis, the given point, 

 viz. the mast-head, being ahove the centre of gravity of the 

 body. But if it be heloiu, as suppose by the projection of 

 the mast beneath, the ellipse ijk I is described, in which 

 the diameter of the orbital circle k i is equal to the minor 

 axis. The other axis is the distance between the two 

 positions which the given point occupies, at the approaching 

 and departing mid-points of the wave. 



22. The axis of the angular motion is parallel to the ridge 

 of the wave. 



Fig. 7. Letaarepresenttheridgeof the waveadvancinginthe 

 direction shown by the arrow ; A and B two floating bodies. 

 Now, as the solid form of the wave may be supposed to be 

 generated by the motion of the trochoidal surface curve 

 along the straight line of the ridge, the water is level at 

 every point of the diameter 6 c of the body A drawn 

 parallel to a a ; the force of buoyancy therefore affects the 

 parts on either side of this line, whence 6 c is the axis ol 

 angular motion. Similarly, in the cubic body B, the line 

 d e passing parallel to a a through the centre of gravity of 

 the body, is the axis of the angular motion of B. For the 

 sake of distinction call this axis of motion, parallel to the 

 ridge, the resultant or wave axis : and the inclination upon 

 it the wave angle. 



Fig. 8. Ifthebodybean oblongfigure of inconsiderable dimen- 

 sions, as at G, and it be required to refer the angular motion 

 on the wave axis d e to angular motions on the axis i j and 

 g h of the body, in order to determine the oblique position 

 of the body with respect to the ridge of the wave, call ij 

 the pitch axis, g h the rolling axis. Let the whole angle of 

 inclination on the wave axis be 9 ; the required inclinations 

 on g h and i j, or the rolling and pitch angles respectively 

 equal to 4> and ij/ ; then, by a well-known theorem of 

 spherics cos = cos ^ cos ij/. 



If a plummet be susjjended from the mast, the angle 

 between the mast line and plummet line is equal to the 

 wave angle. 



23. The greatest angle of inclination of the body is at the 

 mid point of the wave, and is less than half a right angle. 

 In the figure 5, paragraph 20, let h be the mid-point on the 

 circle corresponding to the position e on the wave. Now, it 

 is a pro})erty of tliis curve that a tangent at the point e is 

 parallel to a chord d c of the rolling circle, and this chord 



