Sir William Thomson on Vortex Statics. 107 



in the neighbourhood of vortex-core ; but as different portions 

 of liquid may successively come into the neighbourhood of the 

 core, and pass away again, while the core always remains 

 essentially of the same substance, it is more proper to limit 

 the substantive term a vortex as in the definition I have given. 



22. Definition I. — The circulation of a vortex is the circu- 

 lation [Y. M. § 60 (a)] in any endless circuit once round its 

 core. Whatever varied configurations a vortex may take, 

 whether on account of its own unsteadiness (§1 above), or on 

 account of disturbances by other vortices, or by solids im- 

 mersed in the liquid, or by the solid boundary of the liquid 

 (if the liquid is not infinite), its " circulation " remains un- 

 changed [Y. M. § 59, Prop. (1)]. The circulation of a 

 vortex is sometimes called its cyclic constant. 



Definition II. — An axial line through a fluid moving ro- 

 tationally, is a line (straight or curved) whose direction at 

 every point coincides with the axis of molecular rotation 

 through that point [Y. M. § 59 (2)]. 



Every axial line in a vortex is essentially a closed curve, 

 being of course wholly without a vortex. 



23. Definition III. — A closed section of a vortex is any 

 section of its core cutting normally the axial line through every 

 point of it. Divide any closed section of a vortex into smaller 

 areas ; the axial lines through the borders of these areas form 

 what are called vortex-tubes. I shall call (after Helmholtz) 

 a vortex-filament any portion of a vortex bounded by a vortex- 

 tube (not necessarily infinitesimal). Of course a complete 

 vortex may be called therefore a vortex-filament ; but it is 

 generally convenient to apply this term only to a part of a 

 vortex as just now defined. The boundary of a complete 

 vortex satisfies the definition of a vortex-tube. 



A complete vortex-tube is essentially endless. In a vortex- 

 filament infinitely small in all diameters of cross sections 

 " rotation " varies [Y. M. § 60 (e)] from point to point of 

 the length of the filament, and from time to time, inversely as 

 the area of the cross section. The product of the area of the 

 cross section into the rotation is equal to the circulation or 

 cyclic constant of the filament. 



24. Yorticity will be used to designate in a general way the 

 distribution of molecular rotation in the matter of a vortex. 

 Thus, if we imagine a vortex divided into a number of in- 

 finitely thin vortex-filaments, the vorticity will be completely 

 given when the volume of each filament and its circulation, or 

 cyclic constant, are given ; but the shapes and positions of 

 the filaments must also be given, in order that not only the 

 vorticity, but its distribution, can be regarded as given, 



i 2 



