OF THE POSITIONS OF EQUILIBRIUM. 351 



But the writers on mensuration have demonstrated, that the area of 



the common parabola, is equal to two thirds of its circumscribing 



rectangle, or equal to four thirds of the rectangle described upon the 



axis and the ordinate ; according to this principle therefore, we have 



a'zzi^DcXAC, and a"~%vnXmn. 



By the equation to the curve, it is 



therefore, by division, we obtain 



& 

 DKrr ; 



P 

 but according to the construction, we have 



P 



and by the property of the right angled triangle, it is 



K E 2 -f K m> E m 3 ; that is, E w 2 = 

 therefore, by extracting the square root, we shall have 



P 



and from the similar triangles KETW. and vrm, we get 



Ts.m : Km : : nn : VH; 



and this, by substituting the analytical equivalents, becomes 

 z 



: z : : y : v H ; 

 P ' 

 consequently, by working out the analogy, we have 



py 



VHT 



Hence then, by substituting the respective literal representatives, 

 for the quantities DC, AC and VH, mn, the preceding values of d and 

 a", become 



a'z 4 b, and a"= 



Therefore, let these values of a' and a" be substituted instead of 

 them in the equation (276), and we shall obtain 



= 



If we suppose the axis of the parabola to be vertical, and its base 

 or double ordinate horizontal ; then the points m and D coincide with 



