CHAPTER VII. 



VORTEX MOTION. 



142. OUR investigations have thus far 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 von Helmholtz*; other and simpler proofs of some 

 of his theorems were afterwards given by Lord Kelvin in the paper 

 on vortex motion already cited in Chapter ill. 



We shall, throughout this Chapter, use the symbols f, 77, f to 

 denote, as in Chap, in., the components of the instantaneous 

 angular velocity of a fluid element, viz. 



dw dv\ du dw . dv du\ . 



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 ,,.. 



~~f^ ~~T ........................ \ ) 



V ? 



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

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



* &quot;Ueber Integrate der hydrodynamischen Gleichungen welche den Wirbel- 

 bewegungen entsprechen,&quot; Crelle, t. lv. (1858); Ges. Alh., t. i., p. 101. 



