OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 379 



and stands at right angles to the plane A KB. Consequently, since 

 the vessel is supposed to be inclined around the longer axis, it 

 follows, that the intersection of the planes which we have supposed to 

 be projected into the point P, will be parallel to the axis round which 

 the vessel is supposed to revolve in passing from one position to 

 another. 



But by the laws of hydrostatics, since the whole weight of the 

 vessel is considered to be precisely the same, however much it may 

 be deflected from the upright and quiescent position ; it follows from 

 hence, that the volume which becomes immersed below the water's 

 surface in consequence of the inclination, is equal in magnitude to 

 that which is elevated above it by the same cause, and consequently, 

 the position of the line which is represented by the point P, will depend 

 entirely upon the form of the sides DE and CF. 



Now, in a ship, of which the breadth is continually altering from 

 the head to the stern, and in no regular proportion expressible by 

 geometrical laws, it is manifest, that the place of the point P, repre- 

 senting the line in which the water's surface intersects the vessel in 

 the upright and inclined positions, must be practically determined by 

 some method of approximation, dependent upon the ordinates in the 

 vertical and horizontal sections into which the ship is supposed to be 

 divided. 



By similar modes of approximation, the other quantities necessary 

 for the solution may also be ascertained ; but in ships of war and of 

 burden, constructed after the forms which they generally assume at 

 sea, the calculations necessary for the purpose are unavoidably prolix 

 and troublesome ; and after all, they must depend for their accuracy 

 entirely upon the skill and address of the persons by whom the requi- 

 site ordinates are measured and registered, according to the different 

 parts of the vessel to which they particularly belong ; for if a very 

 nice and accurate arrangement be not preserved with regard to the 

 magnitudes and places of the several ordinates, it is easy to be per- 

 ceived, that the results may come out very wide of the truth, and must 

 therefore necessarily vitiate the whole process. 



In our diagram, the lines DC and EF, through which the principal 

 and secondary sections of the water pass, are supposed to bisect each 

 other, and consequently, the point P must occur at the middle of them 

 both ; in which case its position is known ; but the careful and atten- 

 tive reader will easily perceive, that this can very seldom happen, 

 unless the extreme sides of the zone which limits the angle of the ship's 

 inclination, are equally inclined to, and similarly situated in respect 

 of the extremities of the intersecting lines DC and EF. 



