Prof. Helmholtz on Integrals expressing Vortex -motion. 493 

 5. du du . du 



f. dv dv ^dv 



^dw dw , „div 



At the end of the time dt, the projections of the distance be- 

 tween the two elements of fluid, which at the beginning of dt 

 limited the line qe, have values which by the aid of (3) may be 

 thus written : — 



e^-h{u ] -u)dt=e(^+^dt^, 



er] + (v 1 — v)dt = e\rj+ kJ dt), 

 eZ+{w 1 -w)dt=e(z+ 8 ^ dt). 



The left-hand sides of these equations give the projections of the 

 new position of the joining line qe, the right-hand the projec- 

 tions of the new velocity of rotation, multiplied by the constant 

 factor e. It follows from these equations that the joining line 

 between the elements, which at the commencement of dt limited 

 the portion qe of the instantaneous axis, also after the lapse of 

 dt coincides with the altered axis of rotation. 



If we call vortex- line a line whose direction coincides every- 

 where with the instantaneous axis of rotation of the there- 

 situated element of fluid as above described, we can enunciate 

 the above theorem in the following manner : — Each vortex-line 

 remains continually composed of the same elements of fluid, and 

 swims forward with them in the fluid. 



The rectangular components of the angular velocity vary di- 

 rectly as the projections of the portion qe of the axis of rotation ; 

 it follows from this that the magnitude of the resultant angular 

 velocity in a defined element varies in the same proportion as the 

 distance between this and its neighbour along the axis of rotation. 



Conceive that vortex-lines are drawn through every point in 

 the circumference of any indefinitely small surface ; there will 

 thus be set apart in the fluid a filament of indefinitely small section 

 which we shall call vortex -filament. The volume of a portion of 

 such a filament bounded by two given fluid elements, which (by 

 the preceding propositions) remains filled by the same element of 

 fluid, must in the motion remain constant, and its section must 

 therefore vary inversely as its length. Hence the last theorem 

 may be stated as follows : — The product of the section and the an- 



