358 OF THE POSITIONS OF EQUILIBRIUM. 



Now, the values of x and z as we have determined them by calcu- 

 lation, are respectively equal to 25.19 and 3.75 inches ; therefore, by 

 substitution, equation (279) becomes 



16 2 -j- 4X3.75* 2X25.19 __ 3X40 3.75 2 

 2X16 ~5~~ ~~5~~ ~16" 



from which, by transposition, we have 

 16 2 -f 4X3.75* 2X25.19 3x40 3.75 2 



2 X 16 





445. Here then, it is manifest, that one of the conditions of equi- 

 librium is satisfied, viz. that in which the line which passes through the 

 centres of gravity of the whole section and the immersed part of it, is 

 perpendicular to the surface of the fluid; we have therefore in the 

 next place, to inquire if the areas of the whole section and the im- 

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

 to that of the solid. Now, we have seen, equation (277), that 







vy+42 2 



but by the nature of the parabola, b zn \/ ap ; hence it is 



and we have elsewhere seen, that the value of y y is 



_ 



consequently, by substitution, we get 



therefore, by expunging the common term \/p, and converting to an 

 analogy, we get a^a : x^/x : : s' : s, 



and this, by substituting the given value of a, and the computed value 

 of x, is 



80/10~: 25.19V2539 : : 1 : 0.5. 



From this it appears, that the second condition of equilibrium is 

 also satisfied ; we may therefore conclude, that the positions in which 

 we have represented the body are the true ones ; but we may further 

 observe, that by altering the specific gravity of the body, other posi- 



tions may be exhibited, provided that the expression shall never 

 exceed fl - j* in the common or Apollonian parabola. 



