OF THE POSITIONS OF EQUILIBRIUM. 



353 



consequently, by multiplication, we finally obtain 

 a; =: 40X0.7778 = 31. 112 inches. 



442. Therefore, the position of equilibrium corresponding to the 

 above value of a?, is as represented 



in the annexed diagram, where 

 AB is the base or double ordinate 

 of the parabolic section, DC its 

 axis; FH the water line, or double 

 ordinate of the immersed portion 

 FDII, DE the corresponding ab- 

 scissa, and IK the horizontal sur- 

 face of the fluid. 



That the condition is satisfied, in which the centres of gravity of 

 the whole and the immersed part are situated in the same vertical, is 

 manifest from the circumstances of the case ; and that the other con- 

 dition is satisfied, in which the areas of the whole and the immersed 

 part, are to each other, as the specific gravities of the fluid and the 

 solid, will appear from the following calculation. 



Since the parabolas ADB and FDII, are similar to one another, 

 having the same parameter and being situated about the same axis ; 

 it follows, that 



a^ a : x \/ x : : a : a" ; 



but by the question, a is equal to 40 inches, and by the foregoing 

 computation, we have found that & = 3 1.1 12 inches; therefore, we get 



SOv/To": 31.112t/3nT2 : : 1000 : 686, 



which satisfies the other condition of equilibrium, from which we infer, 

 that if the specific gravity of the solid be taken within proper limits, 

 the preceding diagram exhibits a position of floating. 



443. The equation (278) was obtained on the supposition, that the 

 axis of the parabola is vertical and the points D and m coincident, in 

 which case the quantity z vanishes entirely from the figure ; but the 

 same result will obtain whether we consider the points D and m to be 

 coincident or not, as will appear from what follows. 



By the property of the parabola, that the distance of any point of 

 the curve from the focus, is equal to the perpendicular distance 

 between that point and the directrix, it follows, that mp (see Jig. 

 art. 439) is equal to the sum of DK and DP taken jointly ; that is, 



VOL. I. 



