40 PROCEEDINGS OF THE AMERICAN ACADEMY 



Combining the results obtained at these orifices, grouping those 

 made at the same time or under like circumstances, and giving to 

 each a weight proportioned to the number of observations from which 

 it is obtained, we have the general results contained in the foregoing 

 table. 



These average heights are represented in Fig. 1, Plate No. 3. In 

 this Plate and in Plate No. 4, the horizontal lines represent the sur- 

 face of the stream ; the ordinates are the heights expressed in full 

 scale or in actual distances above and below this surface ; and the 

 abscissas are mean velocities, in which one foot per second is expressed 

 by one twenty-fourth of a foot. 



Taking a general view of these average heights, we find five results 

 are positive and six are negative ; and the general average height of 

 all, or the sum of the products of each height multiplied by its num- 

 ber of observations, divided by the whole number of observations, 

 gives for the 1440 observations, the height of the surface of water in 

 the still-boxes, 0.0007 ft. above that of the surface of the middle of 

 the stream. 



Examining more in detail, we find that, with mean velocities up to 

 four feet per second, the heights of the surfaces in the still-boxes and 

 those at the middle of the stream coincide, within the practicable 

 limits of measurement. With velocities from five feet to nearly nine 

 feet per second, the heights of surfaces in the still-boxes are both 

 above and below those of the middle of the stream by measurable 

 quantities. 



If we now assume, for the purpose of comparison, that, if there is 

 any real difference in these heights dependent upon the velocity, it 

 will vary with the square of the velocity ; and assuming also that the 

 incessant fluctuations of the surface of the stream vary in height with 

 the square of the velocity of the stream, it will be found by plotting 

 these heights with their respective velocities, and giving to each a 

 weight corresponding with the number of observations made in deter- 

 mining it, that they will be represented by a line expressing the 

 height of the surface of the still-boxes above that of the stream by 

 0.000035 of the square of the velocities of the stream ; or a little 

 more than two-tenths of one per cent of the heads which would pro- 

 duce these velocities, and but twelve per cent of the extent of the 

 incessant fluctuations in height of the surface of the stream. 



This result proves that, with these orifices cut with care in pine 

 planks, having their passages at right angles with the side of the 

 trough, and having their edges so nearly in the plane of the side of 



