354 OF THE POSITIONS OF EQUILIBRIUM. 



Z 2 



now, we have already seen that D K is expressed by , and by the 

 nature of the curve, we have DP =r \p ; therefore, by addition, it is 



But since by the property of the parabola, the parameter of any 

 diameter, is equal to four times the distance between the vertex of 

 that diameter and the focus, we have 



P 

 and by the equation to the curve, it is 



consequently, by extracting the square root, we obtain 



P 



Let this value of y be substituted instead of it in equation (277), 

 and we shall obtain 



VP 

 but by the equation to the curve, we have 



therefore, by substitution, we shall get 



VP 

 and multiplying both sides by ^/p, we obtain 



from which, by squaring both sides, we get 



which is the identical expression, obtained on the supposition of a 

 coincidence between the points D and m ; consequently, the value of 

 x must be the same in both cases, and the position of floating depend- 

 ing upon the specific gravity must also be the same. 



Now, by the construction we have seen, that the triangles KEW and 

 wng are similar to one another; hence we get 

 Em : EK : : gn : wn, 



