OF IMMERSED RECTILINEAR FIGURES. 39 



P= the pressure on the entire parallelogram A B c D, 



xrn, the perpendicular depth of the centre of gravity of the 



figure ABCFE, when the side AB is horizontal, 



and y sn, the perpendicular depth, when the side BC is horizontal. 

 Then, because the sides AE and CF are given quantities, it follows, 

 that DE and DF are also given, and consequently, AC or cm, and cb 

 or dm are given ; therefore, the perpendicular pressure on the triangle 

 EDF can easily be ascertained. 



Now, A a is manifestly equal to the difference between A D and a D, 

 and by the construction an is equal to one third of D E ; therefore, by 

 restoring the analytical representatives, we have 



cm = d = l il'. 



Again c b is equal to the difference between c D and D b ; but D b by 

 the construction, is equal to one third of DF ; hence, by restoring the 

 analytical symbols, we shall obtain 



dm $=:b ft. 



But, according to the writers on mensuration, the area of the 

 triangle EDF is equal to half the product of the base DF by the per- 

 pendicular DE ; that is 



JZ'X/3 = jr|3; 



consequently, if we suppose the plane to be perpendicularly immersed 

 in the fluid, while the side AB is coincident with its surface ; then, the 

 pressure on the triangle EDF becomes 



p = %pl's(3l-l'). 



Now, the pressure on the irregular figure ABCFE, is obviously 

 equal to the difference between the pressures on the entire paral- 

 lelogram A B c D, and the triangle EDF; but the pressure on the entire 

 parallelogram, according to equation (8), is 



consequently, by subtraction, the pressure on the figure ABCFE, 

 becomes 



but its area is also equal to the difference between that of the 

 parallelogram and triangle; therefore, we obtain J(2Z (31') for 

 the area of the irregular figure ABCFE; consequently, by division, 

 the perpendicular depth of the centre of gravity below the line AB, 

 becomes 



and if we suppose the fluid in which the plane is immersed to be 

 water, the specific gravity of which i* unity, we finally obtain 



