OF THE POSITIONS OF EQUILIBRIUM* 321 



and by extracting the square root, it is 



2 ) b\l s), 

 and finally, by transposition, we get 



2 a ) # 2 (1 *); (245). 

 the corresponding values of y being 



a?)b\ls}. (246). 

 In order to satisfy the conditions implied in the foregoing equations, 

 it is requisite that the value of s, the specific gravity of the floating 

 solid, should fall between the limits indicated by the following expres- 

 sions, viz. 



-rjz ' ana , j 



now, by the principles of Plane Trigonometry, we have 



b : g \/46 2 a* : : rad. : cos.0, 



which being reduced, gives 



and by involution, we obtain 



Let these values of cos.< and cos 2 .^ be substituted in the above 

 expressions for the limits of 5, and we shall get for the greater limit, 



and for the lesser limit, it is 



__(86 2 a*)X 8 



166 4 



and the arithmetical mean of these two limits, is 

 _ 



Then, if this value of s be substituted instead of it in the equations 

 (245 and 246), we shall obtain for the values of x, as follows, 



2 / 4 / 2 2 \ j.2/1 (,100 ft ) a V /048) 



VOL. I. Y 



