OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 381 



each other, and is therefore coincident with the water's surface, and 

 parallel to the longer axis of the ship. 



471. Since the dimensions of the vessel are supposed to be known, 

 the lines D c and ef will be known ; and from these data, the lines pf 

 and pc are to be assumed by estimation ; but the angle fpc through 

 which the ship is deflected from the upright position, is given by the 

 nature of the particular conditions from which the inclination or de- 

 flexion arises, and consequently, by the rules of Trigonometry, the 

 area of the triangle fpc becomes equal to J/p X jpcsin. 0. 



If therefore, the area of the small circular space fnc be determined 

 by any of the methods of approximation, and added to the area of the 

 triangle fpc, the sum will be the area of the mixed space fpcn, and 

 by proceeding in a similar manner, the area of f'p'c'm will become 

 known ; then, if a mean of these two areas be multiplied by the 

 perpendicular distance pp' , the product will be a near approximation 

 to the solidity of the wedge contained between the planes fpp'f 

 and cppc*. 



And exactly in the same manner, the solid contents of the opposite 

 segment which is elevated by the inclination is to be obtained, and if 

 the aggregate of all the elevated segments be equal to the aggregate 

 of all the depressed ones ; that is, if the entire volume which becomes 

 immersed by the inclination, is equal to that which becomes elevated 

 by the same cause, the point p has been properly determined ; but if 

 they are not equal, the operation must be repeated until they exactly 

 agree, and when this agreement has been obtained, the value of v in 

 equation (282) becomes known. 



472. Now, in order to determine the momentum of stability eli- 

 cited by the ship under the proposed inclination, it is requisite that the 

 product dv in the numerator of the fraction should be completely de- 

 termined ; and for this purpose, the area of the space fpcnf, and 

 the position of its centre of gravity have to be found by approxima- 

 tion, and also, the area of the space f'p'c.'mf y with the position of 

 its centre of gravity. Let the points o and t respectively, denote the 

 positions of those centres, and let the lines or and ts be drawn at 

 right angles to pc andjo'c'; then are pr and p' s the respective dis- 

 tances of the points o and t from the horizontal line pp'. 



Take the arithmetical mean of the two distances pr and p's 9 for 

 the distance between the horizontal line pp', and the centre of gravity 

 of the solid wedge or segment fpcf'p'c'. Find similar distances for 

 all the segments between the head and stern of the vessel, for those 

 which are elevated by the inclination, as well as those which are 



