WATER. 



We now have all the neceflary quantities for making the 



</ n .^ 



calculation thus : multiply — = 1875000 by x = .00022, 



and we have 416.25. To this add — ^ = 4.872, and it makes 



421.122, of which extraft the fquare root, and it is 20.52 ; 



deduft — = 2.207 from this, and it leaves 18.313 inches 

 a 



per fecond for the mean velocity of the water. 



This agrees pretty well with the obfervation of 19 inches, 

 and Dr. Robifon made very nearly the fame refult by a diffe- 

 rent mode of calculation. 



The velocity of 18.313 inches per fecond X 5 gives 

 91.56 feet /icr minute, and again multiplied by 19.25 fquare 

 feet, (the area of the feftion,) gives 176 1.6 cubic feet of 

 water which flow through this canal every minute. 



This example is comparatively eafy, becaufe the table 

 affords the numbers required ; but in I'ome cafes the exaft 

 numbers cannot be found in the table, we Ihall therefore give 

 another example. 



Example 2 Mr. Watt meafured a canal in the neigh- 

 bourhood of Birmingham, which was 18 feet wide at the fur- 

 face of the water, 7 feet wide at the bottom, and 4 feet deep. 

 The water had a declivity of four inches in a mile ; — required 

 the velocity with which the water moved, and the quantity 

 which the canal afforded. 



To have a complete knowledge of the feftion, find the 

 lengtli of each floping fide, thus take the projeftion of the 

 top width over the bottom width on each fide, that is, half 

 the difference between the top and bottom width (18 — 7) 

 — ;- 2 =; 5.5 feet : now the fquare of 5.5 is 30.25, and the 

 fquare of 4 feet the depth is 16, the fum of the two is 

 (30.25 + 16 = ) 46.25 ; and the fquare root of this is 6.8, 

 the length of each Hoping fide. 



Firft, To find the area and the hydraulic mean depth — 

 The mean between the widths of the top and bottom is 

 ( jg 4. 7)-f- 2 12.5 X 4 feet deep = 50 fquare feet for the 

 area of the feftion. To find the circumference which the 

 water touches, add the two floping fides, each 6.8 feet, to 7 

 feet, the width of the bottom, and it makes 20.6 feet. 



The area, 50 fquare feet, divided by 20.6 feet, gives 2.4272 

 feet = 29. 126 inches for the hydrauhc mean depth ; 4 times 

 this is 116.504, which is d, and muft be found in the firft 

 column of the table. The neareft which can there be found 

 is 100 inches. 



Secondly, The fall is 4 inches in the diftance of a mile, 

 = 63360 inches, divide 4 by 63360, and it gives .00006313 

 for 1, the fine of the inclination. 



Thirdly, The value of — , in the fecond column, oppofite 

 a 



to 100 in the firft column, is 2501 000, to which fomething 

 muft be added for the 16.5 inches. To find this quan- 

 tity, take the difference between the adjacent numbers in 

 column two, -viz. 2501000 and 4950000 = 2449000, and, 

 laftly, the difference between 100 and 116.5 — "^-J ' ^^^" 

 fay, as 100 is to 2449000, fo is 16.5 to 404085, which num- 



beris to be added to 2501000, = 2905085, which is — 

 for 1 16.5. 



Fourthly, The value of — , in column third oppofite to 

 a 



100, is 5.272, to which add .0043, as found by a rule of 

 Vol. XXXVIII. 



proportion flmilar to the above, audit gives 5.276, which is 



c^ 



— for 1 16.5. 



Fifthly, Multiply s = .00006313 by 2905085, and it 

 gives 183.395; add —^, or 5.276, as found by the pre- 

 ceding operation, and it gives 188.671 ; and the fquare root 

 of this number is 13.736. 



Sixthly, The value of — , in column fourth, is 2.296 for 

 a 



100, or for 1 16.5 it is 2.297 > deduft this from 13.736, the 

 refult of the laft operation, and we have 1 1.439, which is the 

 velocity of the ftream in inches per fecond, and this x 5 = 

 57.195 feet per minute. To find the quantity, multiply 

 the velocity, 57.19 feet per minute, by 50 fquare feet the 

 area, and we ihall have 285.97 cubic feet, which quantity 

 will flow every minute through this canal. 



The velocity here found is confiderably fmaller than what 

 was obferved by Mr. Watt ; he found the velocity at the 

 furface 17 inches ^^r fecond, and at the bottom 10 inches, 

 the mean velocity we have already calculated at 13.32. 



Dr. Robifon, in the Encyclopedia Britannica, gives a 

 calculation of this fame cafe by Du Buat's formula, which 

 we have given in the article River. He makes the velo- 

 city 11.85 f^^' /"■'■ fecond, which differs fo httle from our 

 computation, that the two theorems may be confidered 

 equally accurate ; but both appear, by Mr. Watt's obferva- 

 tion, to be rather too fmall in very fmall dechvities of 

 rivers and canals. This is not furprifing when we confider, 

 that the experiments, which are the foundation of both 

 thefe formulas, were made on fmall canals ; but for this 

 reafon, we may expeft they will be more accurate when 

 applied to fmaller channels, fuch as mill-courfes, aque- 

 dufts, &c. 



In taking obfervations to apply this method of calculation 

 to praftice, it muft be recoUefted that it always proceeds on 

 the fuppofition, that the canal is of a regular width and 

 depth, and of an uniform (lope throughout. If this is not 

 the cafe, the canal muft be confidered in different portions, 

 and each calculated feparately. We think greater accuracy 

 will be attained by meafuring and carefully levelling lOO 

 yards in which the width and depth are quite regular, than 

 by taking a mile in length, if there are any irregularities 

 in the dimenfions, or in the flope in that diftance. 



On the other hand, the theorem cannot apply at all, unlefs 

 the length of the channel is fuch, that the water in it will 

 arrive at an uniform motion without any acceleration of the 

 motion, as it proceeds down. In fhort and rapidly incHned 

 channels, the water accelerates in confcquence of defcending 

 further down the fall ; but when the canal is long, the ve- 

 locity arrives at a certain point, and then the friftion pre- 

 vents any farther acceleration ; in this cafe, the theorem ap- 

 plies. We fhall not err feniibly in ufing this theorem for 

 canals of 30 yards in length, or Icfs, if the fall is fmall. 



Method of gauging the Water running through clofe Pipes. — 

 Dr. Young's theorem and our table, apply with equal, per- 

 haps greater accuracy, to the cafe of clofe pipes than to open 

 canals. 



All that is neceffary is, to meafure the internal diameter 

 of the pipe in inches, the length of the pipe, and the diffe- 

 rence of the level between the water in the refervoir and the 

 place at which the water is difcharged, and proceed as in 

 the former inftance ; but to render it more clear we ihall 

 give two examples. 



Example I The city of Edinburgh is fupplied with 



p water. 



