164 THIRD REPORT — 1833. 



Thus v= - 0-0469734 + a/ 0-00^2065 + 3041-47 x G, 

 which gives the velocity in metres : or, in English feet, 



?;= - 0-1541131 + \/~0W3751 + 32806-6 x G. 

 When this formula is apphed to pipes, we must take G = :^DK, 



rj I 7 rr 



which is deduced from the equation K = j . When 



it is applied to canals^, we must take G = R I, which is deduced 



from the equation I = :p, R being equal to the mean radius of 



Dubuat on the hydraulic mean depth, and I equal to the sine 

 of inclination in the pipe or canal. M. Prony has drawn up ex- 

 tensive Tables, in which he has compared the observed velo- 

 cities with those which are calculated from the preceding for- 

 mulae, and from those of Dubuat and Girard. In both cases 

 the coincidence of the observed results with the formulae are 

 very i-emarkable, but particularly with the formulae of M. Prony. 

 But the great work of M. Prony is his Noiwelle Architecture 

 Hydraulique, published in the year 1790. This able produc- 

 tion is divided into five sections, viz. Statics, Dynamics, Hydro- 

 statics, Hydrodynamics, and on the physical circumstances 

 that influence the motions of Machines. The chapter on hydro- 

 dynamics is particularly copious and explanatory of the motions 

 of compressible and incompressible fluids in pipes and vessels, 

 on the principle of the parallelism of the fluid filaments, and 

 the efflux of water through different kinds of orifices made in 

 vessels kept constantly full, or permitted to empty themselves ; 

 he details the theory of the clepsydra, and the curves described 

 by spouting fluids ; and having noticed the different phaenomena 

 of the contraction of the fluid vein, and given an account of the ex- 

 periments of Bossut, M. Prony deduces formulae by which the re- 

 sults may be expressed with all the accuracy required in practice. 

 In treating of the impulse and resistance of fluids, M. Prony 

 explains the theory of Don George Juan, which he finds con- 

 formable to the experiments of Smeaton, but to differ very ma- 

 terially from the previously received law of the product of the 

 surfaces by the squares of the velocities, as established by the 

 joint experiments of D'Alembert, Condorcet and Bossut, in the 

 year 1775. The concluding part of the fourth section is de- 

 voted to an examination of the theory of the equilibrium and 

 motion of fluids according to Euler and D'Alembert ; and by a 

 rigorous investigation of the nature of the questions to be de- 

 termined, the whole theory is reduced to two equations only, in 

 narrow pipes, according to the theory of Euler, showing its 

 approximation to the hypothesis of the parallelism of filaments. 



I 



