7 0 Proceedings of the Royal Society 
be so was first discovered and published by Helmholtz. Some- 
times the word vortex is extended to include irrotationally moving | 
liquid circulating round or moving 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 circulation 
[V.M. § 60 (a)] in any endless circuit once round its core. What- 
ever 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 immersed in the liquid, or by the 
solid boundary of the liquid (if the liquid is not infinite), its 
“ circulation ” remains unchanged [V. M. § 59, Prop. (1)]. The 
circulation of a vortex is sometimes called its cyclic constant. 
Definition II . — An axial line through a fluid moving rotation- 
ally, is a line (straight or curved) whose direction at every point 
coincides with the axis of molecular rotation through that point 
[V. 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 “ rota- 
