

OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 387 



sections, as determined by the preceding equation, and thereby draw- 

 ing a plane through all the lines mpn, the plane thus drawn is 

 coincident with the water's surface, and is situated very nearly in 

 its true position. For through the point /<:, draw the line kr meet- 

 ing mn perpendicularly in the point r ; then is krp a right angled 

 triangle in which the angle kpr is given, and by the principles of 

 mensuration, it is manifest, that the solid contained between the 

 planes passing through the lines hki and mpn from head to stern of 

 the vessel, is very nearly equal to the area of the plane passing 

 through hkij drawn into kr the perpendicular thickness of the solid. 

 Now, the solid of which hinm is a section, is obviously equal to 

 the difference of the solids of which fki and hke are sections; hence 

 we have 



m m e AXkr nearly ; 

 consequently, by division, we obtain 



and by the principles of Plane Trigonometry, we get 



kr : kp : : sin. kpr : rad., 

 or by restoring the analytical values, it is 



e 

 : kp : : sm.^> : rad. ; 



and from this, by reducing the proportion and putting radius equal to 

 unity, we obtain 



* _ e _ 



-AXsin.0' (289). 



An equation which is very nearly true for small inclinations, and 

 this being the case, it fully establishes the propriety of the above 

 construction ; if the areas of the planes passing through the lines hki 

 and mpn are equal to one another, the construction as thus effected 

 would be rigorously correct. 



474. In pursuing the construction, it will be necessary, in order to 

 avoid confusion in the lines and letters of reference, to redraw that part 

 of the section which includes the angle of the vessel's inclination, viz. 

 the space contained between the sides of the vessel me, nf and the 

 dotted lines en, inf. We shall not, however, attempt to preserve the 

 due proportion between the several parts of the figure ; this indeed 

 would be troublesome and altogether unnecessary, since it is the prin- 

 ciples of construction only that we mean to illustrate, and not the 

 actual solution of any particular example. 



2c2 



