\\Ver i 

 Vipe*nJ 



of the reservoir, the height or head of water H due to 

 the rod velocity, which, in addition*! tubes, is H = 



for we have already seen iu p. 502, col. 2, that 



Haw** H ~ -_ = 



IB the ft- 



which arise* after this subtraction, must be 

 considered as a declivity to be distributed over the 



HYDRODYNAMICS. 517 



tion of the channel. This line, which may be called 

 d, has been named the Mean radius by Du Buat, and 

 the Hydraulic mean depth by Dr Robison. 



Since the resistances increase as the ratio of the peri- 

 meter of the section, to the area of the section in- 

 creaits increase to the quantity m must be proportion- 

 al to d; and consequently \^mg must be proportional 



. /MS) tr 



should in 



Motion of 



505 



The 





whole length of the pipe. 



In considering theoretically the change which the re- 

 sistances will experience by an increase of velocity, it 

 will appear that they will increase as the squares of the 

 velocity ; for while the impulses on all the little aspe- 

 rities are increased in the proportion of the velocity, 

 the number of impelling particles is also increased in 

 the same proportion. Du Buat therefore supposes the 



resistances equal to , m being a constant quantity 



to be determined by experiment. Now, ifg expresses 

 the velocity acquired by a heavy body at the end of a 



second, will be the accelerating force relative to the 



slope . But from the fundamental axiom, that nhen 



m stream motet uniformly, the remittance it equal to the 



V* - 



accelerating force, we obtain = 2-vn&\'</r=i/ng, 



that is in the same pipe or canal, the product of the 

 velocity by the reciprocal of the square root of the slope 

 is a constant quantity ; and the leading formula for all 



the uniform velocities is V = . 



M. Du Buat now proceeds to examine the preceding 

 equation experimentally, in order to ascertain if Vy' t 

 is actually a constant quantity. After comparing to- 

 gether the results of many accurate experiments, he finds 

 that the values of V v/j, though taken in the same pipe 

 or canal, are not exactly equal, but that they increase a 

 little in proportion as the velocities increase ; and hence 

 he concludes, tkat ike rttutmmett art im m leu ratio titan 

 the tqttare of the telocititt. Hence the term y' * ought 

 to be diminished. The fraction of the slope which Du 

 Buat found to make ^ m g a constant quantity, is </ 1 

 Log. V * + 1.6 in employing the hyperbolic loga- 

 rithm*. Let X represent this fraction ; then we shall 

 have VX = */ mg for the same pipe or canal 



manifest from theoretical considerations, a* well 

 as from direct experiment, that the resistances mutt have 

 a relation to the magnitude of the section of the pipe 

 or canal. As the resistances all arise from the friction 

 of the water upon the sides of the tube or canal, it i. 

 obvious that they must be least in those pipes and canals 

 IB which the section has the greatest ratio to the peri- 

 meter in contact with the water ; that is, the resistance 



Buat found, from many experiments, 

 297. 



Now, since 



0.1)*= 



0.1 



SSJ,M| 



HI 



= 297, we have m = 



297' 



now be expressed by 



V 



we 



-, which is an expression of the vcloci- 



f every particle of water will be in the direct ratio of 937 ( ./,/ QJ \ 



he perimeter of the section, and inversely as the area Hence V= ^==. 



Log.Vi+l.<> 



In cylindrical pipes, the section is the area of a circle ; 

 and the perimeter of the section is the circumference 

 of the circle ; and the quotient arising from dividing 

 the one by the other, is always one-half of the ra- 

 dius ; or one-half of the radius multiplied by the cir- 

 cumference, is always equal to the area of the section. 

 In rectangular and irregular channels, there is still 

 line which, multiplied by the perimeter of the 

 will give an area equal to the area of the sec- which is 



to </d for different channels; and 



Y 



every case be a constant quantity. 



In examining, by experiment, if this was actual!/ 

 the case, Du Buat found, lhAt </mg was neither pro- 

 portional te n/d, nor to any power of d, but that it 

 increased less and less in proportion as t/d increased. 

 In very wide channels */m g becomes sensibly propor- 

 tional to t/r t but in small channels the velocity dimi- 

 nishes much more than the values of </r. Du Buat as- 

 cribes this effect to the viscidity of the water, and he 

 found, that his experiment* were completely represent- 

 ed by diminishing ^d by one-tenth of an inch, that 

 is, by using */d 0.1 instead of t/d; and hence 



-^8 is always a constant quantity, which Du 



to be equal to 



(or making = 243.7) = (Jd 0.1)'. But the re- 



V 1 

 sistances were expressed by , consequently they will 



0.1)* 

 g = >/n~g (vT O.I), and since 



obuin ^^ 



We have also 

 VX = 



897 



ty V for any channel, which, X living a variable quan- 



' I . Du Buat next proceeds to determine. 

 We do not think that our readers will be much in- 

 structed in following our author in his experimental de- 

 termination of X. Upon die supposition that the va- 

 lue X must be sensibly proportional to V when t is 

 great ; that it must always be less than */t ; that it 

 must deviate, from the proportion of ^/*, so much the 

 more that ^t is smaller ; that it must not vanish when 

 the velocity is infinite ; and that it must agree with a 

 series of experiments lor every variety of channel and 

 slope, M. Du Buat found, that these condition- would 

 be fulfilled if we take X = </t Hyp. Log. Vt -4 



M. Buat next supposes, that thrre is a constant por- 

 tion of the accelerating force employed in overcoming 

 the viscidity, and producing the mutual separation of 

 the adjacent filaments ; and he expresses that part of 



the accelerating force by a part of that slope which 



constitutes the whole of it If this were not employed 

 in overcoming a resistance, it would produce a velocity 

 : - really lost ; so that, in reasoning upon this 



