410 OF THE CENTRE OF PRESSURE. 



Therefore, by addition we have d= I -f- , and by taking the fluent 

 of the above equation, we get 



fm ._ 



J - 



3 (a*- ar 



and when a; me?, we shall obtain 



(303). 



The equation as it now stands, is general in reference to a line 

 of which the extremities are both situated below the surface ; but 

 when the upper extremity is coincident with it, then 8 vanishes, in 

 which case c? Z, and our equation becomes 



fm = \L* ( 304 >- 



509. This last form of the expression is too simple to require any 

 illustration ; but the form which it assumes in equation (303), may 

 be expressed in words at length in the following manner. 



RULE. Divide the difference of the cubes of the depths of 

 the extremities of the given line below the surface of the 

 fluid, by the difference of their squares, and two thirds of the 

 quotient will give the distance of the centre of pressure below 

 the surface ; from which, subtract the depth of the upper 

 extremity, and the remainder will show the point in the line 

 where the centre of pressure is situated. 



510. EXAMPLE. Required the position of the centre of pressure in 

 a line of 4 feet in length, when immersed vertically in a fluid, the 



* Now what is here true of a physical line is true also of a plane, which, if it 

 reach the surface of the fluid whose pressure it sustains, will have its centre of 

 pressure at a distance equal to two thirds of its breadth or depth from the upper 

 extremity ; and this holds true also, whatever may be its inclination, its centre of 

 pressure will he distant from the upper edge by two thirds of its surface or breadth. 

 A single force, therefore, applied at that distance, and exactly in the middle of the 

 length of the plane, would hold it at rest. And the same would manifestly be the 

 case, if the rod, in place of being applied longitudinally at a single point, were 

 placed across the plane over the point which indicates the position of the centre of 

 pressure. All that is required in order to procure the equilibrium is, that a 

 sufficient balancing force be applied to that centre ; thus, a sluice or floodgate may 

 be held in its place by the pressure of a single force, applied at one third of its 

 length from its base, and at two thirds of its length below the surface of the fluid. 

 And this suggests the practical importance of placing the beams and hinges of 

 flood and lock-gates at equal distances above and below the centre of pressure, 

 which is at two thirds the depth of the gate. See Problem LXVI. p. 416. 



