HYDRODYNAMICS. 



529 



IMssi tf According to this notation, the formula of Du Buat, 

 in English inches, is 



Pipe* and 



Con ill. 



U = -r^ -^-^ /- -0-3 



Or when reduced to French metres, 

 ^243.79 



0.049359 



About eleven years before the publication of the se- 

 FetmuUof j^nd edition of Du Buat's work, M. Chery obtained an 

 expression of the velocity much more simple than the 

 preceding. He assimilates the resistance of the sides 

 of the pipe or canal to known resistances, which follow 

 the law of the square of the velocity ; and he supposes 



that the ratio of % U* : - - is constant for all cur- 

 rents of the same fluid. Upon this hypothesis, it is 

 sufficient to determine, by experiment, the values of 

 I ", x. nd A for any known current of water, and to 

 deduce from it the general value of U in terms of x. 

 and A belonging to any other current The formula 

 of Chexy, deduced from these principles, is 



I - 



A single experiment only U necessary for determining 

 f, which is an abstract number, or , which is a li- 



quantity. 



The fine researches of Coulomb respecting the resist. 



^ ^ fl^s, were first applied by M. Girard U the 

 discovery of a correct formula for expressing the velo- 

 city of water. He proposed to adopt for the value of 

 the resistance, the product of a constant Quantity, by 

 the sum of the first and second powers of the velocity ; 

 and having determined this constant quantity after 

 12 experiments of Chexy and Du Buat, he obtained a 

 formula, which, as we shall presently see, reiiimjau 

 the experimental velocities as accurately as the more 

 complicated one of Du Buat He txprtsses the resist- 

 ance due to the cohesion by R' x U ; R' being a quan- 

 tity to be obtained from experiment; and sunpoting 

 that the adhesion to the paroi mouilUt, or the film 

 which adheres to the sides of the pipe, of the asperi- 

 ties which are there disseminated, U the same as that 

 which retains the fluid molecules to one another, he 

 makes the resistance due to these asperities equal to 

 R' x L"', so that the sum of the two resistances is 



R'jS (U + U*), which leads to the formula 



R'(l + 



M. Girard assigns, from the experiments of Da 

 Buat, 0.0012181 as the value of R, and his for- 



mula becomes U = 0.5 + </(o.*5+ 8052,54 ^\ 



r j 



or making -f- = -j- = I, and = B, the formula be- 



A ft x 



comet U = 0.5 + J (0.25 + 8052.5* RI). 



In order to obtain a formula for the mean velocity of Motion f 

 fluids, M. Prony found, that an expression of the mean 

 velocity, deduced from the theory of fluids, and com- 

 posed of terms relative to gravity, to the dimensions or 

 figure of the pipe or canal, ought to be equal to a cer- 

 tain function of this mean velocity ; and in determining 

 this function, he observed, that in all the hypotheses re- 

 specting the unknown function of the velocity to which 

 the resistance is proportional that makes the motion 

 uniform, it may always be developed in a series, arran- 

 ged according to the whole powers of the mean veloci- 

 ty, or the variable quantity. That is, 



', &c. 



in which c is a function independent of U, and which, 

 along with the co-efficients , ft, y, &c. must be deter- 

 mined by experiment 



The first term c of this series, is related on the one 

 hand to the inclination which the canal or tube ought 

 to have, in order that the motion may be ready to com- 

 mence ; and on the other hand, to the form and dimen- 

 sions which must be given to the transverse section, in 

 order that the whole fluid which is contained in the ca- 

 nal or pipe may adhere to it The determination of this 

 first term depends on very delicate experiments, which 

 have not been made ; but it is quite certain that, from 

 its extreme sroWlncss, it may be safely neglected. 



The second term U is naturally related to very small 

 velocities ; and as it is known from good experiment*, 

 that the first and second powers of the velocity satisfy 

 all the phenomena included within certain limits, it 



is requisite first to examine if these limits contain the 

 greatest velocities, which are nsremnr to be consider- 

 ed in practice. M. Prony therefore takes the equation 



and he then endeavours to determine the values of the 

 constant Quantities and 3, which may be conformable 

 with the best experiments which have been made on 

 the motion of water in canals. 



In the execution of this task, M. Prony has availed 

 himself of the fine methods for the correction of ano- 

 malies, which M. La Place has applied in his Mecaniqut 

 Ctltttt* for determining the figure of the earth. La 

 Place has given no fewer than three of these methods, 

 the but of which Prony considers as the best 



If we have obtained, for example, a series of experi- 

 mental values of any variable quantity, these values 

 may be connected together by a law, by applying small 

 corrections to each of the experimental results. The 

 equation which expresses this law, may be put under the 

 form 



Zs.+jX, 



where 7. and X are functions of one or more variable 

 quantities, of which we have a certain number of values 

 either directly observed, or calculated from observa- 

 tions. It is then required to assign to the unknown 

 constant quantities and 3 such values, that the phe- 

 nomena may be represented in the best possible man- 

 ner by the preceding equation. 



The explanation of these methods does not belong to 

 the present article ; but in some part of our work, pro- 

 bably under the article I'HVMIS, we shall lay them be. 

 for our readers. 



VOL. XI. PART II. 



JfcttMTM Cfcfr, Psrt I. Lib. iii. 8*0. M. SBi 40. 



Sx 



