OF THE POSITIONS OF EQUILIBRIUM. 349 



of gravity of the immersed and extant parts occur in the same 

 vertical. 



438, We have now to inquire if the second condition of equilibrium 

 be satisfied ; that is, if the area of the whole section and that of the 

 immersed part, are to one another as the specific gravity of the fluid, 

 is to that of the solid. 



Now, the area of the whole section, is 20x20 = 400, and that of 

 the immersed portion, is (1 8.96 -f 11.8)10 = 307.6, or 92.4; and the 

 mean specific gravity is, 0.7693375 or 0.2306625 ; therefore we have 



400 : 307.6 : : 1000 : 769, and 400 : 92.4 : : 1000 : 231 ; 

 consequently, the positions of equilibrium are as exhibited above. 



PROBLEM LX. 



439. A solid homogeneous body, having the section which 

 cuts the axis of motion perpendicularly, in the form of a common 

 or Apollonian parabola, is supposed to float upon a fluid of 

 greater specific gravity than itself: 



It is required to determine the position it assumes when in 

 a state of equilibrium, supposing its base or extreme ordinate 

 to be entirely above the surface of the fluid. 



In the resolution of this problem, we shall have occasion to advert 

 to several properties of the common parabola, a curve which, by 

 reason of its easy construction, and the simplicity of its equation, has 

 been very extensively introduced into mechanical science; and from 

 the frequency of its occurrence, it is presumed, that its chief properties 

 are familiar to and clearly understood by the greatest part of our 

 readers ; so that in tracing the positions of equilibrium, it will not be 

 requisite to demonstrate any of the properties to which we refer, and 

 which, by the nature of the investigation, we are constrained to 

 employ. A 



Let AD B be a common para- 

 bola, representing a transverse 

 section of a solid uniform body> 

 floating at rest upon a fluid 

 whose surface is IK, and let T___ J[j 

 DC be the axis, AB the base or 

 extreme ordinate, (which, by "1 

 the conditions of the problem, 



