CENTRE OF GRAVITY OF PARALLELOGRAMS, &C. 49 



the entire surface ABCD, according to equation (8) under the third 

 problem, becomes 



P = J&/ 2 ; 



and moreover, by equation (12) under the fourth problem, the general 

 expression for the pressure on the triangle ADF, is 



p 4/3 P s sin.0 ; 



but according to the particular case now under consideration, the 

 above expression becomes 



the terms s and sin.0, being each equal to unity, they disappear in 

 the equation. 



Now, according to what we have stated above, the pressure on the 

 trapezoid ABCF, is equal to the difference between the pressure on the 

 entire parallelogram ABCD, and that on the triangle ADF ; that is 



or, by reducing the fractions to a common denominator and collecting 

 the terms, we obtain 



^ j/ = ^(36 2/3). 



By the principles of mensuration, the area of the trapezoid ABCF, 

 is equal to the product that arises, when half the sum of the parallel 

 sides AB andcF, is multiplied by BC the perpendicular distance 

 between them ; that is, 



BC X |(AB + CF)Z= (^ /?), 



and the perpendicular depth of the centre of gravity, is equal to the 

 pressure on the surface, divided by the area of the figure ; conse- 

 quently, we obtain 



jS) ' 



The form of this equation is extremely simple, but it may be arrived 

 at independently of the preceding investigation, by having recourse 

 to equation (20) under Problem 6 ; for according to the conditions of 

 the question, the line of division AF originates at the angle A, and 

 consequently, the perpendicular of the triangle and the length of the 

 parallelogram are equal ; therefore, by putting / instead of /' in equa- 

 tion (20), the above expression immediately obtains. 



Now, by taking the length and breadth of the parallelogram, as 

 given in the preceding example, and the base of the triangle as com- 

 puted by equation (22), we shall obtain, 



3(2 x24 18) 



65. Having thus determined the magnitude of the co-ordinate BS 

 or rm from the equation (25), the magnitude of the corresponding 



VOL. I. E 



