CHAPTER VI. 



VORTEX MOTION. 



126. So far our investigations have been confined for the 

 most part to the case of irrotational motion. We now proceed 

 to the study of rotational or vortex motion. This subject was 

 first investigated by Helmholtz, in Crelle s Journal, 1858; other 

 and simpler proofs of some of his theorems were afterwards given 

 by Thomson in the paper on vortex motion already cited in 

 Chapter in. 



A line drawn from point to point so that its direction is every 

 where that of the instantaneous axis of rotation of the fluid is 

 called a vortex-line. The differential equations of the system of 

 vortex-lines are 



dx dy dz 



T = ~&amp;lt;? = ~? 



where f, 77, f have, as throughout this chapter, the meanings 

 assigned in Art. 38. 



If through every point of a small closed curve we draw the 

 corresponding vortex-line, we obtain a tube, which we call a 

 vortex-tube. The fluid contained within such a tube constitutes 

 what is called a vortex-filament, or simply a vortex. 



Kinematical Theorems. 



127. Let ABC, A B C be any two circuits drawn on the 

 surface of a vortex-tube and embracing it, and let AA be a 



102 



