(1)
as A. Clebsch has shown, from purely analytical considerations (Crelle, lvi.); and then
| ξ = ½ | d(ψ, m) | , η = ½ | d(ψ, m) | , ζ = ½ | d(ψ, m) | , |
| d(y, z) | d(z, x) | d(x, y) |
(2)
and
| ξ | dψ | + η | dψ | + ζ | dψ | = 0, ξ | dm | + η | dm | + ζ | dm | = 0, |
| dx | dy | dz | dx | dy | dz |
(3)
so that, at any instant, the surfaces over which ψ and m are constant intersect in the vortex lines.
Putting
| H − | dφ | − m | dψ | = K, |
| dt | dt |