20 PHYSICAL GEOGRAPHY OF 



With all these precautions errors and discrepancies were unavoidable, and it 

 was only by industriously multiplying observations that their effects could be 

 adequately neuti-alized. 



The first experiment was made on the top of the flood of May 8th, 1849, 

 when the water stood 31tVo feet upon the bar at Wheeling. The observations 

 were continued frequently as the water fell, so as to obtain experimental results 

 for numerous stages, from a flood of 31^ feet down to a depth of 2/(r feet on the 

 bar. 



The value of the discharge was determined by multiplying the area of the 

 reduced prism of the channel corresponding with each stage, by the observed 

 velocity — both in feet — and correcting by De Prony's formula for the difference 

 between the surface and the mean velocity of the stream. But to ascertain the 

 volume of water discharged at every stage of the river, it was necessary to have 

 the means of deducing from the experimental results, corresponding with given 

 depths on the bar, the volumes which must pass down at every intermediate 

 stage. For this purpose it was essential to construct an empirical formula, 

 which should agree with all the correct results obtained from actual measure- 

 ment, and thus permit the interpolation of the quantities due to the intermediate 

 depths. 



The following equation fulfils that purpose: in which, 



d represents the reduced depth of the river, in feet, in the section upon 



which the velocities were measured ; and, 

 D is the corresponding discharge per hour, in cubic feet, when the reduced 

 depth is d. Then, 



1,083,000 d^ _ 10,000 d' = D. 

 or the number of cubic feet hourly discharged when the depth is d. 



This formula agrees satisfactorily with the observed discharges for all the 

 reduced depths at which observations werei taken ; and within those limits may 

 be used in preference to the observations themselves, and to interpolate for 

 heights intermediate between the observations. But, having a maximum value, 

 it would not be admissible to extend its application very far beyond the limits 

 for which it was constructed There is, in fact, no maximum to the discharge, 

 while the value of d increases. 



The following table exhibits, side by side, the quantities discharged, as deter- 

 mined by this formula, and those calculated for the same depths from the obser- 

 vations. 



The first column represents the depth on the bar at Wheeling ; the second, 

 the corresponding reduced depth at the place of observation, or the value of <Z; 

 the third, the observed velocity of the float; the fourth, the calculated discharge, 

 or the value of X>; the fifth is the observed, or measured discharge. 



The average width of the river in the section where the velocities were deter- 

 mined was 1,066 feet, when the depth on the bar was 5/oV feet, and of course 

 varied irregularly with the depth. 



