OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 393 



480. It now only remains to calculate the measure of stability, when 

 the vessel is deflected from the upright position through an angle of 30 

 degrees, and for this purpose we must again refer to the original 

 diagram, where, on the supposition that it is correctly constructed, 

 the lines ki and kh are to be carefully measured in each of the sec- 

 tions, on the same scale with the original dimensions; then, if the 

 lines fi and he be drawn, the areas of the triangular spaces fki and 

 like, can easily be determined from the two sides and the included 

 angle ; and if a series of perpendicular ordinates be measured on the 

 lines fn and he, the areas of the curvilinear spaces/#i and hye may 

 from thence be found, which being added to the triangles fki and 

 hke, the sums will be the whole areas of the compound spaces fki x 

 and hkey. 



Pursuing a similar process throughout the 34 vertical sections, we 

 shall at last arrive at the magnitudes of the volumes which are con- 

 tained between the planes passing through kf, ki and kh, he, a 

 knowledge of these volumes being necessary to determine the posi- 

 tion of the point p. 



It is presumed that it will be sufficient for the exemplification of the 

 rules, to exhibit the calculation of one of the spaces fk ix ; to perform 

 the operation for the whole series, would be a very tedious and at the 

 same time a superfluous proceeding; and for this reason, that the 

 constructions and calculations founded on them, for inferring the 

 results in any one of the sections, are similar to those required for 

 obtaining the corresponding results in any other section ; and this 

 being the case, the representation of one process will suffice for all 

 the rest. 



481. But to proceed, the line ki being taken in the compasses, and 

 applied to an accurate scale of the proper dimensions, it is found to 

 indicate 22.6 feet, and according to the table of ordinates, the line fk 

 is 21.58 feet; and moreover, according to the conditions of the 

 problem, the angle of inclination, or that contained between the lines 

 kf and ki, is equal to 30 degrees, of which the natural sine is j, radius 

 being unity; consequently, if a denote the area of the triangle fki, 

 we have by the principles of mensuration, 



a=|(22.58x21.6)=: 121.927 square feet. 



Now, according to the principles of Plane Trigonometry, the line/i 

 is expressed by the equation 



; V/22.6H-21.58 2 2x22.6X21.58xcos.30zz: 11.55 feet. 



