HIVER. 



tion ; or what (lope, and what magnitude of current, was 

 necelTary for producing any required fupply. Thus, a 

 fmall aqueduct, which was carried to Paris in the beginning 

 of the laft century, on a plan prefented to the academy, 

 and, with fome alteration, approved of by that learned 

 body, was found, when completed, to yield very little more 

 than half the quantity of water which they had computed. 

 A fimilar circumftanee happened at Edinburgh, when that 

 city was fupplied by water, under the direction of Dcfa- 

 guliers ; the quantity of water actually furnilhed being only 

 about one fixth of the quantity which he had computed it 

 would be, and but one-eleventh of what Maclaurin had 

 eftimated it at from the fame plan. 



Nothing can (hew more clearly the inadequacy of the 

 •theory, as it exilled at that time, than that the firft mathe- 

 maticians of their age, and we might almoft add of any age, 

 (hould differ fo widely from each other ; the eltimate of the 

 one being but about one-half of the other, and the neareft 

 to the truth not agreeing with the actual fupply by five parts 

 out of fix. 



It required but a few inftances of this kind to point out 

 the neceliity of a more precife and definite theory ; and the 

 fubject was accordingly foon after undertaken by Miche- 

 lotti at Turin, the abb6 Boffut at Paris, and by the che- 

 valier du Buat ; of whom the latter is generally admitted to 

 have met with the molt complete fuccefs. Michelotti made 

 a great number of experiments, both on the motion of water 

 in pipes and in open canals. They were performed at the 

 government expence, and nothing was fpared to render 

 them complete. A tower of fine mafonry was built, to 

 ferve as a veffel from which the waters were to iffue, through 

 holes of various fizes, and under various preffures, from 

 5 feet to 22 feet. The water was received into bafons con- 

 structed of mafonry, and accurately lined with ftucco, and 

 of various forms and declivities. Thefe experiments on the 

 expence of water through pipes are, of all that have yet 

 been made, the mod numerous and exact, and may be ap- 

 pealed to on every occafion. Thofe made on open canals 

 arc (till more numerous, and are no doubt equally accurate ; 

 but they have not been fo contrived as to be fo generally 

 ufeful, being moltly very unlike the important cafes which 

 will occur in practice ; and they feem to have been contrived 

 chiefly with a view of overturning or el'cablilhing certain re- 

 ceived hydraulic principles of that time. The experiments 

 of Boflut are alio very numerous, and of both kinds, viz. 

 on pipes and canals ; fome particulars of which will be 

 found under our article DISCHARGE of Fluids. But thofe 

 of the chevalier du Buat are the molt conclufive, and his 

 theory of rivers the molt perfect of any with which we are 

 yet acquainted. A few of the leading principles of this 

 author's theory will be found in the fubfequent part of this 

 article. 



It is certain that the motii pen llrcams muft, in 



fome refpects, refemble that of bodies Hiding down inclined 

 planes, perfectly pohlhed ; and that they would accelerate 

 continually, wen- they sot oblfructed : but they arc ob- 

 (trnci ntly move uniformly. Tins can only 



arife I quilibrium li ttween the forces which promote 



their delcent and thofe which oppofe it. Hence M. Buat 

 Hun llfs leading propelition, was. 



I. " When water flows uniformly in any channel or bed, 

 the accelerating force, which obliges it to move, is equal 

 to th' II the refiftances which it meets with, whether 



arifmg from its own vifcidity, or from the friction of its 

 bed." 



From this proportion, ingenioufly combined with the re- 



fult of his own and BofTut's experiments, he then draws thefe 

 fundamental propositions, viz. 



2. " The motion of livers depends entirely on the (lope 

 of their furfaces. 



3. " Since the velocity of the water depends wholly upon 

 the dope of the furface, or of the pipe through which it is 

 conveyed, it follows that the fame pipe will be iufceptible 

 of different velocities, which it will preferve uniform to any 

 diilance, according as it has different degrees of inclinations ; 

 and each inclination of a pipe, of given diameter, has a cer- 

 tain velocity peculiar to itfelf, which will be maintained 

 uniform to any diltance whatever. But this velocity 

 changes continually, according to a certain function of its 

 inclination for all degrees between its vertical and horizontal 

 politions." 



It is obvious that, confidering the number of caufes that 

 may give rile to inequalities in the motion of water, whether 

 in pipes or canals, it would have been vain to attempt the 

 determination of the function above-mentioned from theory 

 only : the refults of the feveral experiments were, therefore, 

 examined with the moft fcrupulous attention, and pene- 

 trating ingenuity, and from which at length the author de- 

 rived the following theorems, viz. 



Let V be the velocity of the ftream, meafured by the 



inches it moves over in a fecond ; R a conftant quantity, 



viz. the quotient obtained by dividing the area of the tranf- 



verfe fection of the ilream, expreffed in fquarc inches, by 



the boundary or periphery of that fection, minus the 



breadth of the ltrcam, expreffed alfo in inches, viz. R = 



to h 



-. r ; where <u> is the mean width of the fection, h the 



b + 2 h 



mean height or depth, and b the breadth at bottom. 



The line R is called by du Buat the radius, and by Dr. 

 Robifon the hydraulic mean depth. 



Laftly, let S be the denominator of a fraction, which ex- 

 preffes the (lope, the numerator being unity ; that is, let it 

 be the quotient obtained by dividing the length of the 

 ftream, fuppofing it extended in a llraight line, by the dif- 

 ference of level of its two extremities ; or, which is nearly 

 the fame, let it be the co-tangent of the inclination or 

 (lope. 



Then the general formula expreffing the velocity V, fup- 

 pofed uniform, is, 



V = 



3°7 



v'R- 



^/S-ih-log. (S++S) 



<J R — Vo, or 



WS-I1 



307 



V = V R 



i h. log. (S + ;■) 



But when R and S arc both very great, then, 





= - R (7-t 



h. log. S 



— -,'„) nearly. 



Hence it follows, that the (lope remaining the fame, the 

 velocities are as R, or as the area ol the fection divided by 

 its perimeter, minus the breadth of the river at the furface, 



very nearly ; for the y are as \' R — ,', ; and when the 

 river is large, the s / R may be ufed without any fcnliblc 

 error. 



Again, if R is fo fmall, that > R — Tl) = o, or R = -,'„, 

 the velocity will be nothing, which agrees very well wit) 

 experiments; for in a cyhndric tube R --- 1 the radius: 

 the radius, therefore, is only two-tenths, fo that the tube 

 is nearly capillary, and the fluid will not flow through it. 

 S f 7 " Tlv 



