HYDRODYNAMICS. 



527 



Hence the discharge is 1107.5 Scotch pints, which 

 differs so widely from 593.3, the quantity actually de. 

 livered, that there must have been some unknown ob- 

 struction in the pipe. 



The cast iron main, five inches in diameter, which 

 is Laid from Comiston to Edinburgh, was always very 

 defective in its delivery. Though its length is only 

 13,518 feet, and the height of the fountain above the 

 point of delivery 88 feet yet it yields only ten cubic 

 leet, or 1^7.7 Scotch pints per minute. 



EXAMPLE IV Mr Watt found, from very careful 

 measurements, that a canal in his neighbourhood, 

 which was 18 feet wide at the surface, 7 feet wide at 

 the bottom. 4 feet deep, and had a declivity of 4 inches 

 in a mile, moved with a velocity of 17 inches per se- 

 cond at the surface, 14 inches in the middle, and 10 at 

 the bottom ; the mean velocity being 1 '(.3. 



Now, since the sloping side of a canal corresponding 



1 ** 7 

 to 4 feet deep, and = 5$ of projection, is 6.8 



feet, we have for the perimeter of the section in con- 

 tact with the water, 6.S + 7 + 6 8 = 20.6. The area 



18^-7 

 of the section will therefore be 4 X - =50 square 



feet Hence <f = - v = 2.*:72, or 29.1*6. The 



logarithm corret ponding to this in Table III. is 3 2 1 1 1 7, 

 and the value of the negative quantity in col. 5. i 1.589. 

 Now, since the slope i* 4 inches, or , of a foot in a 

 mile, we have * = 15 840, and the corresponding lo- 

 garithm in the table 2.0828O. Hence, 



From S.SIM 7 



Subtract 2.08280 



And there remains 1.12837 = Log* of 13.439 inches 

 Subtract the negative quantity 1.589 



And there remains 1 1.850 inches, 



the velocity of the canal per second required, which 

 differs considerably from 13.33, the mean velocity ob- 

 served by Mr Watt 



In the two first examples, the reader will observe, that 

 the formula errs in defect Dr Kobison considers it as 

 most correct in small canals where it is most needed, such 

 as hi mill courses and other derivations for working ma- 

 chinery. From several comparisons with direct observa- 

 tion, he propose* to substitute in place of the expression 

 Hyp. Log. vT+T6 the expression 2{ Com. Log. 

 N 'j + 1.6, which be considers both M more simple and 

 more accurate. 



np- Instead of the part of the numerator Hyp. Log, 

 Jjjjjj, /* ~4- 1.6, Dr Young proposes to substitute 0.85* ii, 

 Md by which is nearly the same for moderate velocities. He 



- l).(^ + ~ -.OOl) 



proposes also V = 307 



and since. > may be substituted without much inac- 



1 t 



Curacy in place of ' *, the term - ^ will become 



1.6*' 



which may be determined without logarithms. 



Hence the whole formula will become 



i being the length of the pipe, A the height of the whole 

 head oC water, and d the diameter of the pipe. la this 



^w -.< V I 



l + 45d -Motion or 



formula $ = iy . The formula may be applied to Water in 



Pipes and 

 1 Canal*. 



river*, by taking - as the sine of their inclination. w v ~-' 



M. Langsdorf has proposed to substitute 482 in place 

 of the number 478 used by Du Buat in his formula 

 V=v/478A in French inches, which gives V=^/509A 

 in English inches. 



When the pipe is bent in one or more places, the ef- change* 

 feet of the bending may be found by adding into one proposed by 

 sum r the squares of the sines thus, Langviorf. 



^\ 



09^ soooy 



509 dh 



h (. 

 Or more simply, 



M, 



which is Langsdorf s formula reduced to English mea- 

 sure. 



M. Eytelwein conceives the head oF water to be di- Fl 

 vi<le<l into tu-o part-, one of which is employed in pro- b'!J*g_. 

 ducing velocity, while the other is employed in over- telireiji. 

 coming the resistances to which it is exposed. He con- 

 siders the height employed in overcoming the resist, 

 ance* to be directly "as the length of the pipe, and as the 

 circumference of the section, or as the diameter of the 

 pipe, and inversely as the area of the section, or as the 

 square of the diameter ; that is, on the whole, inversely 

 at the diameter. This height too, roust, like the resist* 

 ance arising firom friction, vary as the square of the ve- 

 locity. Hence if /"denote the height due to the fric- 

 tion, > the diameter of the pipe, and a a constant quan- 

 tity, we shall have 



/s:Vt if, andV*=^-f. 

 i a I 



But the height employed in overcoming the friction, 

 Corresponds to the difference between the actual veloci- 



\ 

 ty and the actual height, that is./= A -j- where 6 is 



the co-efficient for finding the velocity from the height 

 Hence we have 



V = 



and V 



= /_Ej 

 V aPt 



a if I 



Now Boat found b to be 6.6, and a 6* was found to be 

 0.02 1 1, particularly when the velocity is between 6 

 and 24 inches per second. Hence we have 

 _43.6|A 



-0.02I7T+T 



Or, what is considered more accurate, 





If the pipe is bent, the velocity thus found must be Application 

 diminished by taking the product of its square, multi- of H )'<' 

 plied by the sum of the sines of the several angles of *""'* {ot ' 

 flexure, and then by 0.0038. This will give the de- n " lU- 

 gree of prMsure employed in overcoming the resist- 

 ance occasioned by the angles; and subtracting this 

 height from the height corresponding to the velocity, 

 we may thence find the corrected velocity. 



In applying this formula to Example 2. in p. 526, re. 

 lative to the 4 inch pipe, which supplies Edinburgh 

 with water, we have A =51 feet. ) =. 0.375 of a foot, 



