OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 385 



In the present instance, the vertical sections being all different, 

 both in form and magnitude, the water's surface intersecting the 

 vessel in the plane passing through the line mn when the vessel is 

 inclined, will so divide the areas of the several sections, that although 

 the space fpn may not be equal to mpe in any one of them, yet the 

 immersed volume corresponding to all the spaces fpn, estimated 

 from the head to the stern of the ship, shall be equal to the volume 

 corresponding to all the emerged spaces mpe estimated in the same 

 manner. 



Let ef, the breadth of the section at the water line, be bisected in 

 the point k by the vertical line dc, and suppose a plane to pass 

 through dc from head to stern of the vessel, such a plane will divide 

 the vessel into two parts that are equal and symmetrical, and it will 

 pass through the point k in all the parallel vertical sections made 

 throughout the whole length. 



But it is easily shown, that at whatever distance kp from the 

 middle point k, the plane of floatation in the inclined position, inter- 

 sects the primary line ef in one of the vertical sections, it will 

 intersect the corresponding line in all the other sections at the same 

 distance from the middle point; that is, the distance kp will be the 

 same in all the parallel sections, (the same lines and letters of 

 reference being understood to belong to each ;) for according to the 

 conditions of the problem, the revolution of the vessel is supposed to 

 be made about the principal longitudinal axis, and consequently, the 

 intersection of the two planes passing through the lines ef and mn, 

 must be parallel to the axis of motion, and therefore parallel to the 

 line drawn through the point k in all the sections, estimated from 

 head to stern of the ship. 



We have now to determine the distance kp at which the inter- 

 section takes place ; and for this purpose we must consider, that 

 according to the given conditions, whatever may be the position of 

 the point p in all the sections, if lines mn are drawn through those 

 points, making with ef, an angle equal to the given angle of inclina- 

 tion ; then it is manifest, that the same plane will pass through all 

 the lines mn that occur betwixt the head and stern of the vessel. 



It is therefore required to determine, at what distance kp from the 

 middle points k, the plane of floatation corresponding to the inclined 

 position of the vessel must pass, so as to cut off a volume on the 

 depressed side fpn equal to that which rises above the water on the 

 side mpe. 



In each of the parallel vertical sections, let the common line hi 

 VOL. i. 2 c 



