52 Canon ^loseley on the steady Flow of a Liquid. 



rig. 4 represents a vertical section through the axis of the 

 same current^ the line D C being the depth of the current in the 

 centre, and being divided as the central line (vertically) of fig. 1 

 is divided. Lines are drawn parallel to the surface through 

 these points of division ; and the velocities at the corresponding 

 depths, as shown in fig. 1, are set off twelve times along each of 

 these parallel lines. These points being joined laterally, the cor- 

 responding curves represent the positions into which the vertical 

 filament of fluid, which coincides at any instant with D C, would be 

 brought at the end of 1", 2", 3", &c. from the time of its starting. 

 The point of maximum velocity, as shown in fig. 4, is not at the 

 surface, but at 0"03 metre from it. This explains the lip-like form 

 of the curve as it approaches the surface. If we imagine a line 

 of particles of camphor to be made instantaneously to fill the 

 line D C, they will arrange themselves, after 1", 2", ?>", &c., in 

 the corresponding lines of the figure. If instead of these par- 

 ticles being made to fill a vertical line, they had been made to 

 occupy a vertical cross plane through D C, they would, after 

 these given periods of time, have arranged themselves succes- 

 sively in curved surfaces, of which the current-lines of fig. 3 re- 

 present the intersections with the surface of the current, and 

 those of fig. 4 the intersections with a vertical plane through 

 the axis. 



By equations (52) and (53) we obtain for the case of this 

 channel 



zJi=:l-2456-'^"69.r^ iJ2=l-245e-^-23i/, 



where v^ is the velocity of the current measured across from the 

 surface at a distance x from the centre, and v^ is the velocity 

 at a vertical depth y from the surface measured from the surface. 

 Beneath the observed velocities in fig. 1 are given the theoretical 

 ones as determined by these formulae. 



If i" represent any number of seconds and each formula be 

 multiplied by if, {iv^) will represent an ordinate of a current-line 

 of the surface of the liquid, corresponding to the abscissa £c, i" 

 after it has left the position AB ; and {tv^) will represent a similar 

 ordinate of the current-line of the depth corresponding to the 

 abscissa y. 



Putting (tv^)=Zi, and {tVo)=Zf, we have 



^, = (1-2450 6--769^, ^2=(l'S^50e-^'23i/, 



which are the equations to the current-lines of the surface and 

 the depth respectively. 



The Value of fi. 

 In the preceding investigation /x is taken to represent the 

 statical resistance per unit of surface to the motion of one film 



