Researches in Vortex Motion. 



333 



question of the stability of these simple motions or of their resultant, 

 it is clear that if we superpose the two we get another state of 

 steady motion in which we have vortex filaments in tbe shape of 

 helices lying on concentric cylindric surfaces. The problem to be 

 considered is whether it is possible to conceive a similar superposi- 

 tion of two motions in the case of any vortex aggregate whose 

 motions are symmetric about an axis. 



The general conditions for the existence of such systems 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 function 

 -»/r in the usual manner. The velocity along a parallel of latitude is 

 given by v = f(ty)p where p is the distance of the point from the 

 straight or polar axis. The function satisfies an equation of the 

 form (when expressed in polar co-ordinates) 



dr* + r 2 dd* r 2 dO p J df ' 



where F and / are both functions of ty. The case F uniform, and 

 / oc yjr is treated more fully. If / = \yp-ja where a is the radius of 

 the aggregate, 



^ =A { j2 (?) _ 5 JW } sinie - 



The most striking and remarkable fact brought out is that as X 

 increases we get a periodic system of families of aggregates. The 

 members of each family differ from one another in the number of 

 layers and equatorial axes they possess. According to the number of 

 independent axes they are called singlets, doublets, triplets, &c, in 

 contradistinction to more or less fortuitous or arbitrary compounds 

 of the former, which are considered later and called monads, dyads, 

 triads, &c. Of these families two are investigated more in detail 

 than the others, both because they are specially interesting in their 

 properties and because they serve as limiting cases between the 

 different series. In one family (the X 2 family) all the members 

 remain at rest in the surrounding fluid. In the other (the \ family) 

 a distinguishing feature, common to all the members, is that the 

 stream lines and the vortex lines are coincident. 



The parameter X gives the total angular pitch of the stream lines 

 on the outer current sheet, although in aggregates with more than 

 one equatorial axis these lines are not one continuous line. The first 

 aggregates — with X<5'7637 (the first X 2 value) — behave abnormally. 

 Beyond these we get successive series, in one set of which the 

 velocity of translation is in the same direction as the polar motion of 

 the central nucleus, in the alternate set the velocity is opposite, and 



2 b 2 



