390 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 



RULE 2. y#* = (S-f4p-|-2Q)Xir, 



where x, y, r and 5 denote as in rule 1st; p the sum of the 2nd, 

 4th, 6th, 8th, &c. ordinates, and Q the sum of the 3rd, 5th, 7th, 9th, 

 &c. (the last of the series excepted). 



RULES. / X =(S + 2p + 3g)X|r, 



here again, x, y, r and S denote as in the preceding cases ; P the sum 

 of the 4th, 7th, 10th, 13th, &c. ordinates (the last excepted), and Q 

 the sum of the 2nd, 3rd, 5th, 6th, 8th, 9th, &c. 



With respect to the applicability of the above rules, it may be 

 observed, that the first approximates to the fluent, whatever may be 

 the number of the given ordinates, and the second only requires that 

 the number of ordinates shall be odd. But in order to apply the 

 third rule, it is a necessary condition, that the number of given 

 ordinates shall be some number in the series 4, 7, 10, 13, 16, &c. ; 

 that is, the number of ordinates must be some multiple of 3 increased 

 by unity. In every case, however, the approximate fluent can be 

 obtained, either from the second or third rule considered separately, 

 or from both taken conjointly. 



477. But to return from this short digression, we may remark, that 

 the position of the point G, which marks the centre of effort, or the 

 centre of gravity of the whole vessel, depends partly on the equipment 

 and construction, and partly upon the distribution of the loading and 

 ballast; which circumstances, therefore, determine G#, the distance 

 between the centre of effort and the centre of buoyancy when the 

 vessel is upright. 



These several conditions having been determined, the remaining 

 part of the construction, limiting the measure of the vessel's stability, 

 may be effected as follows. 



Through g the centre of buoyancy, or the centre of gravity of the 

 immersed volume, draw gt parallel to mn, and make gt to BR (see 

 the subsidiary figure), as the volume immersed in consequence of the 

 inclination, is to the whole immersed volume induced by the weight 

 of the vessel ; through G the centre of effort, draw GZ parallel and 05 

 perpendicular to gt, and through t draw tm parallel to GS, and meet- 

 ing the axis cd in M ; then is M the metacentre, and GZ the measure 

 of the vessel's stability when inclined from the upright position through 

 the angle fpn or gGs. 



The principles of the preceding construction are general, and can 

 be applied in all cases, whatever may be the figure of the vessel, or 



