14 OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 



But the equation, in its present form, supposes the specific gravity 

 of the fluid to be expressed by unity, which only takes place in the 

 case of water ; in order, therefore, to generalize the formula, we must 

 introduce the symbol which denotes the specific gravity ; hence, 

 we obtain 



pi= |/ 2 s sin. -\- Isd; 

 or by collecting the terms, we get 



p ls(\ Zsin.^>-|-c?). (5). 



27. This is the general form of the equation, on the supposition 

 that the perpendicular depth of the upper extremity of the line is 

 given; it however assumes a different form, when the depth of the 

 lower extremity is known ; for by Plane Trigonometry, we have 



as above 



b erz:sin.0, 



and by subtraction, we obtain 



ec bc be; that is, dznp / sin. ; 



therefore, the perpendicular depth of the centre of gravity is 



3=iD \ I sin. cf), 



and consequently, the general expression for the pressure becomes 

 p=ls(o \ I sin. 0). (6). 



28. Therefore, the practical rule for each of these cases, when 

 expressed in words at length, is as follows : 



1. When the perpendicular depth of the upper end is given (5). 



RULE. To half the length of the given line drawn into the 

 natural sine of the angle of inclination, add the depth of the 

 upper extremity ; then., multiply the sum by the length of the 

 line, drawn into the specific gravity of the fluid, and the pro- 

 duct will give the pressure sought. 



2. When the perpendicular depth of the lower end is given (6). 



RULE. From the perpendicular depth of the lower extre- 

 mity, subtract half the length of the given line drawn into the 

 natural sine of the angle of inclination ; then, multiply the 

 remainder by the length of the line, drawn into the specific 

 gravity of the fluid, for the pressure sought. 



29. EXAMPLE 2. A physical line, whose length is 56 feet, is immersed 

 in a cistern of water, in such a manner that its upper extremity is at 

 the distance of 9 feet below the surface, and its direction making with 

 the horizon an angle of 58 degrees ; required the relative pressure on 

 the line, the water being in a state of quiescence ? 



The natural sine of 58 degrees, according to the Trigonometrical 

 Tables, is .84805 ; therefore by the rule, we have 



28 X .84805 + 9 = 32.7454, the perpendicular 



