WATER. 



the fedion r— tT x 4 j = 50 fquare feet, by the breadth 

 of the bottom and length of the floping fides added toge- 



1 ther ; whence we have 



50 



or 29.13 inches : and the fall 



8 X 29.13) = 

 mean proportional; -i-°ths of which is 13.9, 



20.6 

 in two miles being 8 inches, _ we ^ have 

 15.26 for th , . 



agreeing nearly with Mr. Watt's obfervation. Profeflbr Ro 

 bifon has deduced from Buat's elaborate theorems 12.568 

 inches for the velocity, which is confiderably lefs accurate. 



For another example we may take the river Po, wliich 

 falls one foot in two miles, where its mean depth is 29 feet, 

 and its velocity is obferved to be about 55 inches in a fecond. 

 Our rule gives 58, which is perhaps as near as the degree 

 of accuracy of the data will allow. 



On the whole, we have ample reafon to be fatisfied with 

 the unexpe&ed coincidence of fo fimple a theorem with ob- 

 fervation ; and in order to find the velocity of a river from 

 its fall, or the fall from its velocity, we have only to recol- 

 left that the velocity in inches per fecond is |4ths of a mean 

 proportional between the hydraulic mean depth and the fall 

 in two Englifh miles in inches. This is, however, only true 

 of a ftraight river flowing through an equable channel. 



For the Hope of the banks of a river or canal, Mr. Eytel- 

 wein recommends, that the breadth at the bottom fhould be 

 j-ds of the depth, and at the furface Vds ; the banks will 

 then be in general capable of retaining their form. The 

 area of fuch a feftion, is twice the fquare of the depth, and 

 the hydraulic mean depth fds of the aftual depth. 



M. Du Buat's Rule. — In our article River, we have 

 given the theorem of M. Du Buat for calculating the motion 

 of water in a river or other regular channel, or through 

 pipes. It has been obferved by the late Dr. Robifon, 

 that the comparifon of the chevalier Du Buat's calculations 

 with his experiments is very fatisfaftory ; that it exhibits a 

 beautiful fpecimen of the means of exprefling the general 

 refult of an extenfive feries of obfervations in an analytical 

 formula ; and that it does honour to the penetration, fltill, 

 and addrefs of M. Du Buat, and of M. De St. Honore, 

 who affifted him in the conftruftion of his expreffions. 



Dr. Young's Rule Dr. Young julHy remarks, in an ex- 

 cellent paper in the Philofophical Tranfaftions for 1808, that 

 the form of Du Buat's exprcflions is not fo convenient for 

 praAice as they might have been rendered ; and are liable to 

 great objeftions, in particular cafes : for when the pipe is ex- 

 tremely narrow, or extremely long, they become completely 

 erroneous. Dr. Young has, therefore, fubftituted for the 

 formulse of M. Du Buat others of a totally different nature ; 

 and he profefTes to have followed Du Buat only, in his general 

 mode of confidering a part of the prefTure, or of the height 

 of a given fall, as employed in overcoming the friftion 

 of the pipe, through which the water flows out of it ; a 

 principle which, if not of his original invention, was cer- 

 tainly firil publifhed by him, and reduced into a practicable 

 form. We find Mr. Smeaton ufed it in conllrudting his 

 MS. tables. By comparing the experiments which Du 

 Buat has collefted, with fome of Gerftner's, and fome of his 

 own. Dr. Young difcovered a formula, which appears to 

 agree fully as well as Du Buat's, with the experiments from 

 which his rules were deduced, and at the fame time accords 

 better with Gerftner's experiments ; and which formula ex- 

 tends to all the extreme cafes with equal accuracy. It feems 

 to reprefent more fimply the aftual operation of the forces 

 concerned ; and it is dircft in its application to practice, 

 without the neceffity of any fucceflive approximations. 



He began by examining the velocity of the water dif- 

 charged through pipes of a given diameter, with different 



degrees of prefTure ; and found that the friaion could not 

 be reprefented by any fingle power of the velocity, although 

 it frequently approached to the proportion of that power 

 of the velocity, of which the exponent is i .8 ; but that it ap- 

 peared to confill of two parts, the one varying fimply as the 

 velocity, the other as its fquare. The proportion of thefe 

 parts to each other mnft, however, be confidered as dif- 

 ferent, in pipes of different diameters ; the firfl part being 

 lefs perceptible in very large pipes, or in rivers, but be- 

 coming greater than the fecond in very minute tubes ; while 

 the fecond alfo becomes greater, for each given portion of 

 the internal furface of the pipe, as the diameter is dimi- 

 nifhed. 



If, with Dr. Young, we exprefs all the meafures in 

 Englifli inches, calhng the height employed in over- 

 coming the friftion /, the velocity in a fecond v, the 

 diameter of the pipe d, and its length /; we may make 



+ 2 c -J V : for it is obvious, that the fric- 



tion muft be direftly as the length of the pipe ; and finee 

 the preffure is proportional to the area of the feftion, and 

 the furface producing the fridtion to its circumference or 

 diameter, the relative magnitude of the friftion mud alfo be 

 inverfely as the diameter, or nearly fo, as Du Buat has 

 juftly obferved. 



nr 

 We fhall then find, that a muft be .0000001 (413 + -^ 



a 



_ 144° 

 d+ 12.8 



f <)0O dd 





180 



I 



^ d 



and c muft be .0000001 



+ :7^('°'^ + 



13.21 



+ 



•0563 \> 

 dd )/ 



\dd+ 1136 ^ ^d\ -" ' d 



Hence it is not difficult to calculate the velocity for any 

 given pipe, open canal or river, with any given column 

 of water : for the height required for producing the velo- 



city, including fridlion, is, according to Du Buat, 



510 



or rather, as it appears from almoft all the experiments 



586 



586 



or h 



and the whole height h 

 ■a I 



= &+i6>' 



I 



586. 

 and alfo 



which the doftor compared, 



is, therefore, equal to f -\- 



-I — V ; and afTuming b 



d 



aliummg e = -— j-, we nave 



i>= ^ [h h -\- e''') — c ; which is a general theorem. 



In order to adapt this formula tp the cafe of rivers, we 



muft make / {the length) infinite ; by which h becomes 



al -T- d-y .00171 

 ' -j- 2e-u = bh \ whence, 



/' 



niS.bh = 



ds 



X -r = 



s being the^nir of the in- 



a i a 



clination of the water's furface, and d =z 4. times the hy- 

 draulic mean depth. The hydraulic mean depth is the area 

 of the fedion of the moving water, divided by as much of 

 the circumference of that area, as the water touches. 

 / [ads + e') — c 



fince e is here = — -u = 



And 

 and in mod 



a a 



rivers, v becomes nearly ^/ ( 20000 <//). 



Another ufeful rule by Dr. Young, is to find the fuper- 

 ficial velocity of the water in a river by adding to the mean 



velocity 



