OF THE POSITIONS OF EQUILIBRIUM. 307 



389. The above modification, however, does not strictly accord with 

 the conditions of the problem ; we must therefore inquire, whether the 

 just principles of equilibrium do not depend upon some other element, 

 such as the specific gravity. Now, we have already stated, that in 

 order to have the values of x and y real positive quantities, it is neces- 

 sary that 



s , cos s .0(46 a -a 9 ) 



should be less than - prs - 

 s 4kb 9 



and for a similar reason 



s . 



must be greater than 



And if the specific gravity of the fluid be denoted by unity, as is 

 the case with water, then the specific gravity of the floating body must 

 lie between the limits 



cos 9 ^(4& g a a ) and cos.ft V 46* a 9 

 4& 2 b 



The specific gravity of the floating body, as we have proposed it in 

 the question, is 686, that of water being denoted by 1000; conse- 

 quently, when the specific gravity of water is expressed by unity, that 

 of the solid is 0.686; let us therefore try if this number lies between 

 the above limits ; for which purpose, we must substitute 28 for b, 20 

 for a, and 0.93406 for cos.0 ; then we shall have as follows. 



0.93406 2 (4X28 ? 20 2 ) 

 For the greater limit we have s = - ;. OQa - - 0.761 



4X ~ 



nearly. 



It is therefore manifest, that the specific gravity of the floating 

 body, as we have employed it, is less than the greater limit, and con- 

 sequently properly chosen with regard to it, and we have next to 

 inquire if it exceeds the lesser limit ; for which purpose, it is 



28 



Here then it is obvious, that the lesser limit exceeds the given 

 specific gravity ; and from this we infer, that without the modification 

 specified above, the body will not fulfil the conditions of the problem 

 in any other position than that in which its base is parallel to the 

 surface of the fluid ; but if the specific gravity of the floating body 

 fall between the numbers 0.761 and 0.745, all other things remaining, 

 then the prism, besides the situation of equilibrium in which its base 

 is parallel to the surface of the fluid, may have two others, in both of 



x2 



