370 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 



from which, by equating the products of the extreme and mean terms, 

 we get 



aXnz~ a'd; 



therefore, by division, we have 



a'd 



nz . 



a 



But by the principles of Plane Trigonometry, it is 



rad. : 3 : : sin.0 : nr, 

 which being reduced, gives 



wrzzd sin.^, 



and according to the construction of the figure, it is manifest that rz 

 or G s, is equal to the difference between nz and nr, the two quantities 

 whose values have just been determined ; consequently, by subtrac- 

 tion, we have 



dd . . 

 X ~~a asm * ' (282). 



456. The equation which we have just investigated has reference 

 only to a particular case of the general problem, viz. that in which the 

 vertical transverse sections, throughout the whole length of the body, 

 are equal and similar figures ; this condition, although it is a restric- 

 tion upon the general applicability of our result, yet it allows an im- 

 mense latitude, for the figures of bodies whose parallel transverse 

 sections are equal and similar areas are very numerous ; and if we 

 substitute the magnitude of the whole immersed volume, and that 

 of the volume which becomes immersed in consequence of the inclina- 

 tion, instead of the areas of the respective sections, the above 

 equation becomes general, because its form and the manner of 

 combining the terms admit of no change. 



The expression consists of five members on one side, one of which, 

 that is, the angle of deflexion, must always be a given datum, or it 

 must be directly assignable from the circumstances of the case, and 

 the others must all be determined by means of the given dimensions, 

 and other particulars dependent upon the figure of the section ; but 

 the method of applying the formula, and the whole operation neces- 

 sary for its reduction, will be sufficiently exemplified by the resolution 

 of the following example. 



457. EXAMPLE. A solid homogeneous body, of which the trans- 

 verse parallel sections at right angles to the axis of motion, are 

 equal and similar trapezoids, is placed upon the surface of a fluid in 

 such a manner, that its broadest side is upwards and parallel to the 



