64 REPORT — 1881. 



could not be set up or destroyed by a system of conservative forces ; bnt 

 this was almost all. Helmholtz first ^ave ns clear conceptions in liis 

 well-known paper' referred to above. He introduced tlie idea of vortex- 

 lines (curves whose tangents at any point' coincide in direction with the 

 axis of rotation at that point), and vortex- filaments (the portions of fluid 

 in a tube whose surface is generated by vortex-lines passing through an 

 infinitely small curve). Helmholtz' laws are then proved — (1) Each vortex 

 remains continually composed of the same elements of fluid. (2) The pro- 

 duct of the section at any point of a filament into the magnitude of the 

 rotation at the point is constant for all time and for all points of the 

 filament. Also (3) a vortex-filament must be closed or have its ends on 

 the boundary of the surface. The next part of the paper is devoted to 

 finding expressions for the velocity when the rotations at every point of 

 the fluid are known ; in other words, to finding solutions of the four 

 differential equations.^ 



du dv dio ^ 



dx^ dy'^ Iz~ ' 

 dw dv ar t e 



- — — - = 2i, &c., &c. 

 di/ dz 



which reduce to three on account of the relation 



d'ijdx -f drj/dy + de/dz = 



supposed existing between the given quantities t,, »;, c. The introduction 

 of the functions L, M, N, has been noticed above. It is shown that they 

 are the potential functions of distributions of magnetic matter, whose 

 density at any point is £/27r, ?y/2T, e/27r, and thence that each rotating 

 element of fluid (a) implies in each other element (b) of the same 

 fluid a velocity which follows the same law as the fori;e exerted on 

 a particle of magnetism at (h) by the element of an electric current at (a), 

 in the axis of rotation. 



In the same paper, examples are given of the motion of the fluid due 

 to infinite straight vortices and to circular vortex-filaments. It must be 

 remembered that the results here given refer to the cyclic motion of the 

 fluid as determined by the supposed distribution of magnetic matter, and 

 do not give the most general motion possible. Helmholtz shows that 

 this motion in the case of straight parallel vortices is such that, regarding 

 their strengths as positive or negative masses, according as they rotate in 

 one or the other direction, their centres of gravity remain at rest. When 

 two vortices are of equal and opposite strength they move forward 

 together with constant velocity and at a constant distance. This solves 

 also the case where a single vortex is in a fluid bounded by an infinite 

 plane, to which it is parallel. The results for circular rings are more 

 complicated, and since in nature the section mu^st be finite, an indeter- 

 minateness enters on account of the distribution of rotation within it. 



N. Jonkoffsky, Moscaycr Math. Samml. viii. 



E. Beltrami, ' Sui principi fondamentali dell' idrodinamica,' Mem. di Bologna, i. 

 ii. iii. V. 



The two latter treat systematically the kinematics of the motion. Beltrami, 

 amongst other things, compares the motion of an element of fluid with what it would 

 have been if suddenly solidified, and the loss of energy thereby. 



' Crelle, Iv., translated in Phil. Mag. (4) 33, p. 48.5. 



• Helmholtz employs the opposite sign for \, i\, f. 



