^2 OF HYDRAULICS. 



consequently 111. u'+i-r: A; k being the actual heighf, 



389. Theorem. When a river flows 

 with a uniform niotiorij its velocity varies as 



the square roots of the hydraulic mean depth ^""^ {aib'+d).v'=b^dk, and «'=_!____ 



and of the sine of the inclination conjointly. c^„„, , _, _ . , . . , . 



J J ScHoiioM. The coefficient t IS in this case 8.8, and 



For since the relative weight produces no acceleration, it "''* '^ "early .0211, where the velocity is moderate : but it 



must be exactly balanced by the friction : but the friction is is more accurate to make t)=:50^ (— £L_) ; all the mea- 



as the square of the velocity, therefore the relative weight \ '+50d/ 



must be as tlie square of the velocity, and the velocity as ^"^" ''^'"S ejcpressed in English feet. When the pipe is 



the square root of the relative weight, or of the sine of the ''^"'' ^'^ ""^y '^"'' ^'^^ •'«'Sht employed in*ovcrcoming the 



inclination. And since the friction produced by the bed ailditional resistance, by multiplying the square of the ve- 



of the river is for any given portion of water, as the extent ^°'^''>' ^y ^'^^ ^'"= of «he angle of inflection, and by .0038. 



of the bed directly and inversely as the quantity of water, ^* "^V "'^o ''^^"cs '''^ velocity of a river from the same 



that is inversely as the hydraulic mean depth, the square of f'"''""'*' supposing the pipe to be in train, or so constituted, 



the velocity must vary in the direct ratio of the depth in '^'" ''" velocity is independent of its length ; for mating 



order to produce a given friction, and the velocity must be A=zc+A/, we have ^""^"^^^'^ constant, whatever may be 



as the square root of the mean depth, +5oa 



the magnitude of 2,andif /:::aO ,V=.bo ■/ {kd) ; but(/t)is the 



Scholium. It is found by experience that the mean ve- sine of the inclination, and e being the mean depth, d:=.ie 



locity of a river in a second is nearly nine tenths of a mean and riz y (i ooooAr?) ; or, if we employ .02 1 1 as the value of 



proportional between the hydraulic mean depth, and the at-Si;=^/(828oie); whilethcrule deducedfrom observation 



fall in two English miles. And if this velocity be ex- is equivalent to «:r:v/[BS53*e.) 



pressed in inches and increased by its square root, it will _, ^^ i , i. 



. , ... ,. , 1 1 ■ u J I ■. SQL Iheorem. The hydraulic pressure 



give tne velocity of the surface, if diminished by its square j r ^ 



root, the velocity at the bottom. It appears however that O^ ^ j^t acting directly on a plane surface, 



the velocity increases a little more rapidly than the square SO as to lose its whole progressive motion, is 



root of the fall. The discharge of a were may be found by equal to twice the weight of a column OU 



determining the velocity due to the sum of its height, and ^i , j i- .u i • i .. i 



" . , , the same base, and or the heieht correspond- 



the height corresponding to the velocity with which the ■■ 



water arrives at it. ing tO the Velocity. 



390. Theorem. The square of the velo- ^°' '" '^^ '""" '"'•"''"'^ ^°' ^ ^°'^5' '° f"" *™"5h this 



,„.,,., , , , . height, each particle of the jet would lose its velocity by 



city or a fluid discharged throus^h a pipe va- ^ . ,. , .... u . • .u- .- 



•' ° o r r ti)£ immediate action of gravitation ; but in this time the 



ries d irectly as the height multiplied by the ^^^^ number of particles lose their velocity by the reaction 



diameter, and inversely as the diameter in- of the surface as are contained in a column on the same 



creased by a certain constant fraction of the base, and of twice the height = therefore the effects being 



Ipno-tVi ■ equal, the causes must also be equal. 



392. Theorem. The resistance of a fluid 



The height of the reservoir above the orifice of the pipe , . , 1 • . 1 



■., , ,. ij . . .u 1 J to a body moving through It, is as the square 



may beconsidered as divided into twopartSjtheoneemployed tv^ " "- j a o ' 1 



in overcoming the friction, the other in producing the ve- Oi the velocity. 



locity. Now the whole friction varies directly as the length For the relative motions are nearly the same as in the 



of the pipe, inversely as its hydraulic mean depth, which is impulse of a jet ; and the height of the column varies ac 



one fourth of its diameter, and directly as the square of the the square of the velocity. 



velocity ; or calling the height employed on the friction,/, ScHOiruM. When the impulse is oblique, the resistance 



/=1. v\ a being a constant quantity. But the height tnay be calculated from the laws of the decomposition of 



force ; but the results are not accurate enough to be of any 



employed in producing the velocity v, is 2L,ibeingthepro- use witliout a comparison with experiments. 



per coefficient fw determining the velocity from the height; 393. THEOREM. When the whole of a 



