46 PARALLELOGRAM DIVIDED TO SUSTAIN 



according to the problem, is double the pressure on the triangle adf; 



hence we have 



P 2p i= 6p' = 1089787.23 Ibs. 



61. If the line of division were drawn from one of the lower angles 

 to a point in the immersed length, after the manner represented in the 

 annexed diagram ; then, the equation (22), would assume a different 

 form, as will become manifest from the following investigation. 



From the angle d on da and dc the sides of the 

 parallelogram, set off dn and dt, respectively equal to 

 one third of df and dc, and through the points n and 

 t thus found, draw the straight lines nm and tm \ 

 parallel to dc and df, the base and perpendicular of 

 the triangle fdc, and meeting one another in m, the 

 place of its centre of gravity. 



Produce tm directly forward, meeting a b, the hori- 

 zontal side of the given parallelogram perpendicularly in the point r ; 

 at the point r in the straight line mr, make the angle mrs equal to 

 the angle of the plane's inclination, and draw ms perpendicularly to 

 rs ; then is sm the perpendicular depth of the centre of gravity of the 

 triangle /We. 



Let therefore, the notation of the preceding case be retained, and 

 put x == df; then we have 



am^irmm. I' ^x, and consequently sm di=: (I ^x) sin .0 ; 

 but the area of the triangle fdc is expressed by \b x; therefore, the 

 pressure perpendicular to its surface, is 



now, according to the conditions of the problem, the pressure on the 

 separated triangle is equal to half the pressure on the entire paral- 

 lelogram ; consequently, we obtain 



and this, by expunging the common quantities, becomes 



2 x (3 / x) = 3 l\ 

 or dividing by 2 we get 



x(3l x} = l.5l\ 

 and from this, by separating and transposing the terms, we have 



x* 3lx 1.5 Z 2 . (23). 



If the equations (18) and (23) be compared with one another, it 

 will readily appear, that they are precisely similar in form, but dif- 

 ferent in degree ; the former being an incomplete cubic, wanting the 

 first power of the unknown quantity, and the latter an adfected 

 quadratic, having all its terms. Indeed, the diagrams from which the 



