ATOM. 469 



The co-ordinates of this point are 



x + adcr, y + ftdv, z + ydar ........................... (9), 



and the components of its velocity are 



Consider the first of these components. In virtue of equation (7) we may 

 write it 



du 7 . du _ 7 du j 



du dx , du dy ^ du dz , /ir .v 



or u + -j- -j- da-+ -j- -f- d<r + -j- -3- da- .................. (12), 



dx dcr ay da dz do- 



or u + (d<r ................................................... ( 13 )' 



But this represents the value of the component u of the velocity of the fluid 

 itself at the same point, and the same thing may be proved of the other 

 components. 



Hence the velocity of the second point on the vortex line is identical 

 with that of the fluid at that point. In other words, the vortex line swims 

 along with the fluid, and is always formed of the same row of fluid particles. 

 The vortex line is therefore no mere mathematical symbol, but has a physical 

 existence continuous in time and space. 



By differentiating equations (1) with respect to x, y, and z respectively, 

 and adding the results, we obtain the equation 



^ + ^ + ^ = ................................ (14). 



dx dy dz 



This is an equation of the same form with (6), which expresses the con- 

 dition of flow of a fluid of invariable density. Hence, if we imagine a fluid, 

 quite independent of the original fluid, whose components of velocity are a, ft, y, 

 this imaginary fluid will flow without altering its density. 



Now, consider a closed curve in space, and let vortex lines be drawn in 

 both directions from every point of this curve. These vortex lines will form 

 a tubular surface, which is called a vortex tube or a vortex filament. Since the 

 imaginary fluid flows along the vortex lines without change of density, the 



