HYDRODYNAMICS. 



533 



TABLE IV. 



Containing the observed Measures in TABLE III. in 

 French Inches. 



Pronv next proceeds to investigate single f,, r 

 mula wh,ch w.ll .erve both for canals and pipe,. Th 



reuniting formula is 

 U = -0.046973* + 



When this formula is applied to canals, we must take Mii" n ii ' 



rr r Water in 



G = RI ; I = -^ ; and R = . 

 A X 



When it is to be applied to pipes, we must take 



H' i ? TJ" 

 G = J D J, D= diameter of pipe, and J = 



Pipes and 

 Canals. 



when the pipe discharges itself in water, or J = -^, 



when the pipe discharges itself in air ; as in this case, 

 H"= 0, H' being the height of the head of water above 

 the superior orifice of the pipe, and H" the height of, 

 the head of water above the lower orifice of the pipe. 

 The formula in English feet is, 



U= 0.1541 131+^(0,023751 -f 32806.6 G.) 



On the Relation between Ike Superficial Velocity and the 

 Mean Velocity. 



The formula give by Du Buat for deducing the mean Relation 

 from the superficial velocity was, when reduced to the between the 

 metre, upofcial 



U = (y'V _ 0.08227)* + 0.0067675, 

 which is deduced from the equation 



where W u a constant velocity = 0.0270699 = 1 inch 

 of the old measure, and y' W = 0.16453. 



Although this formula is sufficiently simple, and har- 

 monizes with many of Du Bust's experiments, it is 

 nevertheless incompatible with observation, as it makes 

 the mean velocity I have a finite value, when the su- 

 perficial velocity V U nothing. Now, as Prony has 

 observed, every formula which doe* not make both 

 these velocities vanish at the same time, is evidently 

 erroneous ; and, as it follows front the examination of 

 the experiments, that the ratio between these velocities 

 approaches to equality as they increase, so that at one 

 limit we have V = 0, U = 0, and at another limit V= 

 oo, U = * ; and V = U. 



la order to obtain a formula which should satisfy 

 these conditions, and at the same time be simple, and 

 suited to the nature of the phenomena, M. Prony give* 

 it the form 



*> 



V-l-6 

 which may be put under the form, 



rs - e= s -T - + -r - , and making - - = 

 V U 6 a n it a B 6 a 



and -r =j, and using the values of V and U, given 



in col 2. and 3. of the following Table, he obtained 

 by the methods already mentioned. 



= 4.036 ; ft = 1.280, from which we have 

 a =2.37187; 6 = 3.15312, and 

 U- V ( V + 2 -37187) 

 V+ 8.15312 ' 



a formula which i* not only more commodious, and 

 more easily calculated, but also more conformable with 

 experiments than that of Du Buat. This formula nlfcy 

 be put under the form 



from which we obtain, 



0.0022065-1-3041.470). V = 4, (U-2J 7 187+v'(U-2.37187)+3.15312[J) 



