OF IMMERSED RECTILINEAR FIGURES. 43 



therefore, by analogy, we obtain 

 50 : 19 :: 28 : y = 10.64 feet. 



Here, the whole process of calculating the second co-ordinate, is 

 replaced by the simple analogy above exhibited. 



The example now before us, affords a striking instance of the 

 advantages to be derived from this mode of considering the centre of 

 gravity; in the case of the triangle illustrated under Problem (B), its 

 immediate utility was not so conspicuously displayed ; but we are 

 convinced, that in figures of more difficult and complicated forms, its 

 usefulness will become still more evident. 



In the investigation of the formulae, we have thought it necessary 

 to consider the pressure on the surface whose centre of gravity is 

 sought ; but in the actual application of the resulting equations, the 

 consideration of pressure does not enter; for it is manifest, that 

 besides the dimensions of the figure and constant numbers, no other 

 element is found in the equations, and consequently, the reduction 

 depends upon them alone. 



7. OF EQUAL FLUID PRESSURES ON THE SECTIONS OF A RECTANGULAR 

 PARALLELOGRAM AND THE PERPENDICULAR DEPTHS OF THE CENTRE 

 OF GRAVITY. 



PROBLEM VIII. 



58. A given rectangular parallelogram is immersed in an 

 incompressible and non-elastic fluid, in such a manner, that one 

 of its sides is coincident with the surface, and its plane tending 

 downwards at a given inclination to the horizon : 



It is required to draw a straight line from one of the upper 

 angles to the lower side, so that the pressures on the two 

 parts into which the parallelogram is divided, may be 

 equal to one another. 



Let AED represent a rectangular cistern filled with water, or some 

 other incompressible and non-elastic fluid, 

 of which ABEF is the horizontal surface, 

 and suppose one of the upright sides, as 

 ABCD to be removed, exhibiting the fluid 

 together with the immersed rectangle abed. 



In dc the lower side of the immersed 

 parallelogram, take any point/, and draw 

 af to represent the line of division ; then 

 the triangle adf, and the trapezoid abcf, T> 



are the figures into which the parallelogram is divided, and on which 

 the pressures are equal. 



