48 



S. PERPENDICULAR DEPTH OF THE CENTRE OF GRAVITY Of A PARAL- 

 LELOGRAM DIVIDED INTO TWO PARTS SUSTAINING EQUAL PRESSURES. 



64. With respect to the centre of gravity of the figure a b cf, which 

 remains after the triangle adf, or fdc has been separated from the 

 parallelogram, it is in this particular instance very easily determined ; 

 for, since the area of trapezoids, whose parallel sides and perpendi- 

 cular breadths are equal each to each, are also equal ; it follows, that 

 the centre of gravity must occur in the straight line which bisects the 

 parallel sides ; it is therefore, only necessary to investigate the theorem 

 for calculating one of the co-ordinates, the other being determinable 

 from the circumstance just stated. 



Let ABCF be the trapezoid, having the angles at B and c respec- 

 tively right angles, and of which the position of the centre of gravity 

 is required. 



Produce the side CF directly forward to any convenient length at 

 pleasure, and through the point A, draw the straight line AD parallel 

 to BC, the longer side of the trapezoidal figure, and 

 meeting c F produced perpendicularly in the point D. 



Then, the pressure upon the trapezoid ABCF, is 

 manifestly equal to the difference between the pressure 

 on the parallelogram A BCD, and that upon the triangle 

 ADF, and its area, is also equal to the difference be- 

 tween their areas. Bisect the parallel sides AB and CF 

 in the points a and b, and join a b ; then, according to 

 what lias been demonstrated by the writers on mechanics, the centre 

 of gravity of the trapezoid ABCF occurs in the straight line ai. 



Suppose it to occur at m, and through the point m draw mr and 

 ms respectively parallel to BC and BA, meeting AB and BC perpen- 

 dicularly in the points a and b ; then are rm and sm the co-ordinates, 

 whose intersection determines the position of the point m. 



Put b =z AB, the breadth of the parallelogram A BCD, 

 / AD, or BC, its corresponding length, 



= m r the depth of the point m as referred to the line A B con- 

 sidered to be horizontal, 

 cTzu ms, the depth of the point m as referred to the line BC under 



similar circumstances, 

 and /3~ DF, the base of the triangle ADF. 



Then, by conceiving the plane to be immersed perpendicularly in a 

 fluid whose specific gravity is expressed by unity, the pressure upon 



