OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 17 



But when the lines are differently situated in the fluid, the compa- 

 rison of their relative pressures requires a more particular exemplifi- 

 cation; for which purpose take the following example. 



32. EXAMPLE 3. Two physical straight lines, whose lengths are 

 respectively 18 and 27 feet, are immersed in the same fluid, in such 

 a manner that their upper extremities are just in contact with its 

 surface, and the angles which they make with the horizon are respec- 

 tively equal to 42 and 29 degrees ; what is the pressure on the longer 

 line, supposing that on the shorter to be expressed by the number 

 78.54?. 



If we convert the preceding analogy for the oblique lines of different 

 inclinations into an equation, by making the product of the mean 

 terms equal to the product of the extremes, we shall obtain 



Now, by assimilating the several quantities in this equation to the 

 lines in the foregoing diagram, and according to the conditions of the 

 question, it appears that p' is the required quantity, all the rest being 

 given ; therefore, let both sides of the equation be divided by / 2 sin. 0, 

 and we shall obtain t 



, _ p /* sin. <f 

 / 2 sin.</> ' 



But it is a well-known principle in the arithmetic of sines, that to 

 divide by the sine of any arc, is equivalent to multiplying by the 

 cosecant of that arc ; hence we have 



, 



p'=(p sin. 0' cosec.</>). 



Let therefore the numerical values, as proposed in the example, be 

 substituted for the respective symbols in the above equation, and we 



shall obtain 



.972 



' = (78.54 sin. 29 cosec.42) ; 

 18 2 



now, the natural sine of 29, according to the Trigonometrical Tables, 

 is .48481, and the natural cosecant of 42 is 1.49447 ; therefore, by 

 substitution, we get 



if ?Z!x 78.54 X .48481 X 1.49447 128 nearly ; 



18 2 



consequently the pressures on the inclined lines, are to one another as 

 the numbers 78.54 and 128; but had the inclinations been equal, 

 the comparative pressures would have been as 78.54 to 176.72 very 

 nearly. 



VOL. i. c 



