VOKTEX MOTION. [CHAP. VI. 



where the summations embrace all the vortices present in the 

 fluid. If in these equations we suppose #, OT always to refer 

 to the same vortex, we may write 



dx d sr 



u= Tt v = Tf 



Since ra is constant for the same vortex, the equation (39) 

 is at once integrable with, respect to t } whence 



2 war 2 = const .......................... (40). 



A quantity OT O defined by the equation 



2 



may be called the mean radius of the vortex-rings. The 

 equation (40) shews that this mean radius is constant throughout 

 the motion. 



If we introduce in addition a magnitude x such that 



x Sm-sr 2 = 5)w ^x ..................... (42) , 



it is plain that the position of the circle whose co-ordinates are 

 # , CT O depends only on the strengths and the configuration of 

 the vortices, and not on the position of the- origin of co-ordinates. 

 This circle may be called the circular axis of the whole system 

 of vortex-rings. It remains constant in radius ; and its motion 

 parallel to x is obtained by differentiating (42) , viz. we have 



dx - d-& 



or, by (38) and (41); 



where we have added to the right-hand side a term which 

 vanishes in virtue of (40). 



141. The formulae (11) become, on making the substitu 

 tions (37), 



-0 



M , 7C . , j , . , 



M = - - &amp;lt;sr d% dxdv t 



