II. Researches in vortex motion.—Part III. On spiral or gyrostatic vortex aggregates

The chief part of the following investigation (Sects, i. and iii.) was undertaken with the view of discovering whether it was possible to imagine a kind of vortex motion which would impress a gyrostatic quality which the forms of vortex aggregates hitherto known do not possess. The other part (Sect, ii.) deals with the nongyrostatic vortex aggregates, the discovery of which we owe to Hill, and investigates the conditions under which two or more aggregates may be combined into one. It is shown that it is allowable to suppose one or more concentric shells of vortex aggregates to be applied over a central spherical nucleus, subject to one relation between the radii and the vorticities. In all cases the vorticities must be in opposite directions in alternate shells. The special case when the aggregates are built up of the same vortical matter is considered, and the magnitudes of the radii and the positions of the equatorial axes determined. The cases of motion in a rigid spheroidal shell and of dyad spheroidal aggregates are also considered. The chief part of the paper refers to gyrostatic aggregates. The investigation has brought to light an entirely new system of spiral vortices. The general conditions for the existence of such systems, when the motion is symmetrical about an axis, are determined in Sect, i., and are worked out in more detail for a particular case of spherical aggregate in Sect. iii. It is found that the motion in meridian planes is determined from a certain functionψ in the usual manner. The velocity along a parallel of latitude is given byv =f(ψ) /ρ where p is the distance of the point from the axis. The functionψ , however, does not depend on the differential equation of the ordinary non-spiral type, but is a solution of the equationd 2 ψ /dr 2 + 1/r 2 d 2 ψ /dθ 2 - cotθ /r 2 dψ /dθ =ρ 2 F -f df /dψ , where F andf are both functions ofψ . The case F andfdf/dψ both uniform is briefly treated. It refers to a spiral aggregate with a central solid nucleus, and is not of great interest. The case F uniform and feef∝ψ is treated more fully. If ≡λψ/a wherea is the radius of the aggregateψ = A { J2 (λr/a )- (r2 /a2 ) J2 λ } sin2 θ.