NOTCHES AND WEIRS 143 



computed mean velocity over the whole section of the approach channel. 

 But the distribution of velocity over this section is not at all uniform, the 

 filaments nearer the surface and nearer the centre having, a velocity greater 

 than the mean, depending on the depth, width, and surface condition of 

 the channel. The particles which suffer the least change in their direc- 

 tion of motion on passing the weir are thus those which have a velocity 

 of approach greater than the mean, and since the velocity of approach in 

 these particles has a greater proportional effect in increasing the flow 

 past the weir than in those which approach its plane in a more oblique 

 direction, the effective velocity of approach will be greater than that 

 corresponding to the calculated head h. Most experimenters, therefore, 

 put the effective head as equal to H + <* h, where a has some value greater 

 than unity, and write 



Q = K b (H + a /i)3 = Kb D$. 



Francis, however, allows for the velocity of ap poach by putting H -{- }t 

 and h as his limits of integration in formula (1) of this article, and thus 



gets 



For a suppressed weir Q = 3 -33 b j (ff+ fc) - ftf J 

 With one end contraction Q = 3'33 (b - -1H) I (H + K$ - 

 With two end contractions Q 3'33 (b '2 fl) j (H + h 



In using Francis's formula, care should be taken to apply the correction 

 in this manner. 



2 



The value of h = -z is determined by approximation. Thus Q is 



* 9 



first determined from the measured H, and this value of Q is used to 

 determine v from the known area of the channel. 



The second approximation to the true value of Q, obtained by inserting 

 the value of h thus found, in the expressions given above, is always 

 sufficiently near for all practical purposes. 



s. r a. s i 



Since in this formula we have D 2 = \ (H + h)' 2 h' 1 Y 



.% D= 



Expanding this as a series and omitting all terms containing a small 

 quantity of the second order, we get 



D = H + h | V 5 = H + &. where a = I - | A/-^ 



