OF THE POSITIONS OF EQUILIBRIUM. 303 



denoted by m, subtract and add the quantity denoted by n, 

 and the difference in the one case, and the sum in the other, 

 will give the values ofy corresponding to above values of x. 



385. These are the rules by which the other positions of equilibrium 

 are to be determined ; but it is necessary to remark, that beyond cer- 

 tain limits no equilibrium can obtain. In the first place, in order that 

 the body may float with only one of its angles immersed, it is mani- 

 festly requisite, that the equal sides of the section should each be 

 greater than m -j- n ; and in the second place, in order that x and y 

 may be real positive quantities, the expression J cos 2 .^>(46 2 a 2 ) must 



. tfs s , cos 2 .d>(4 2 a ) 



exceed , or must be less than - r . ,, - . 

 ' 2 



reason of these limitations is obvious from the nature of the 

 quadratic formula (233) and (234), but it will be more satisfactory to 

 show, that unless the data of the question are so constituted as to 

 fulfil these conditions, the rules will fail in determining the positions 

 of equilibrium ; or in other words, there is no other position in which 

 the body will float at rest, but that which is indicated by the equa- 

 tions (231) and (232). 



386. EXAMPLE. The data remaining as in the preceding example, 

 let it be required to determine from thence, whether under the pro- 

 posed conditions, the body can float at rest in any other position than 

 that which we have already assigned for it, by the reduction of the 

 equations (231) and (232), in which the extant side or base of the 

 figure is parallel to the horizon. 



By the principles of Plane Trigonometry, we have 



\ cos.0 n= T V v 7 28 + 1 0) (28 1 0) = !(0.93406) = \ cos.20 55' 29" ; 



consequently, by proceeding according to the rule, we'get 

 m~\ cos.</>(46 2 a s )*zz0.46703 ^4 X28 2 20*=: 24.429 very nearly. 



Again, to determine the value of w, it is 



i cos. 8 0(46 2 a 2 ) =. 0.46703 2 (4 X28 2 20 2 ) = 596.768, 



and for the value of the term, involving the specific gravities, we have 



consequently, by subtraction, we get 



596.768 537.824 = 58.944. 



It therefore appears from the last result, that both the values of x 

 and y are real positive quantities ; consequently, one of the limiting 



