MOST ECONOMICAL FORM OF TRANSVERSE SECTION. 221 



the float more easily visible. Such a rod, so adjusted in 

 length that it sinks nearly to the bed of the stream, gives 

 directly the mean velocity of the whole vertical section in 

 which it floats. (For a complete description of gauging 

 streams, see " Report on the Mississippi.") 



119. Most Economical Form of Transverse Sec- 

 tion. The best form of the transverse section must be that 

 which presents the least resistance to a given quantity of 

 water flowing through the channel. From Art. 103, the 

 resistance of the bed of the stream, in consequence of the 

 adhesion and friction, varies directly as the surface of con- 

 tact, and consequently as the wetted perimeter ;j (Art. 113), 

 and inversely as the area of the transverse section, i. e., the 



v 



resistance of the bed of the stream varies as - In order, 



a 



therefore, to have the least resistance from friction, the 

 form of the section must be that which has the least perim- 

 eter for a given area, i. e., the wetted perimeter^ must be a, 

 minimum for a given area , or the area must be a maximum 

 for a given wetted perimeter. Now, among all figures of 

 the same number of sides, the regular one, and among all 

 the regular ones, the one with the greatest number of sides 

 has the smallest perimeter for a given area. Hence, for 

 closed pipes, the resistance of friction is the smallest when 

 the transverse section is a circle ; but in open channels, the 

 upper surface, being free, 

 or in contact with the air 

 alone, must not be in- 

 cluded in the perimeter. 



A horizontal line DC, 

 passing through the centre 

 of the square AF, divides 



the area and perimeter into two equal parts, and what has 

 been said of the square is true of these halves ; hence, of all 

 rectangular forms of transverse sections, the half square 



