OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 389 



centre of gravity of the volume whose sections are the areas pmre ; 

 then is B R the horizontal distance between the centres of gravity of 

 the volumes that are respectively immersed and emerged, below and 

 above the water's surface, in consequence of the vessel being deflected 

 from the upright position, through an angle of which the magnitude 

 is known. 



475. The solid content of the entire volume immersed, or the quan- 

 tity of water displaced by the immersed part of the vessel, is to be 

 obtained from the areas of the several horizontal sections ; for the 

 ordinates drawn in the several sections being arranged in regular order, 

 after the manner which we have adopted in the preceding table, the area 

 of any section can readily be assigned, by methods of approximation 

 adapted to the particular case, and from these areas the solidity of 

 the immersed volume is to be inferred ; making allowance for the 

 irregularities of the vessel towards the head and stern, if it be at all 

 necessary to take those parts into the account ; in all practical cases, 

 however, they may safely be omitted. 



That part of the immersed volume, comprehended between the keel 

 and the lowest horizontal section, is obtained, by first finding the areas 

 of the several vertical planes, between the keel and the nearest ordi- 

 nates, and from these areas, by means of some appropriate mode of 

 approximation, the magnitude of the part cut off by the lowermost 

 horizontal plane will be determined ; which being added to the solidity 

 of the part contained between the extreme planes, will give the mag- 

 nitude of the immersed volume, or the quantity of fluid displaced. 



476. Referring to the original diagram, it will be observed, that from 

 the areas of the several horizontal sections, made between the keel of 

 the vessel and the plane of floatation, the distance kg, that is, the 

 distance between the water line ef and the centre of buoyancy, or 

 the centre of gravity of the immersed volume, can also be determined 

 by the application of particular approximating rules, and the best 

 with which we are acquainted for this purpose, are those given by 

 Stirling in his " Methodus Differ enlialis" and by Simpson in his 

 " Essays ;" these rules may be expressed in general terms as 

 follow. 



RULE 1. Jyx=i(?\S}Xr, 



where x is the fluxion of the abscissa, y the perpendicular ordinate, 

 expressing a general term or function of x ; r the common distance 

 between the ordinates ; S the sum of the first and last ordinates, and 

 p the sum of the whole series. 



