534 



Prof. W. M. Hicks. 



the aggregate regredes in the fluid as compared with its central 



aggregate. 



Suppose the attempt made to obtain sets of aggregates with greater 

 and greater angular pitch. It will be found that as the external 

 pitch of the stream lines increases the equatorial axis contracts and 

 the surface velocity diminishes. On the outer layers (ring-shaped) 

 the spiral pitch is chiefly produced on the inner side facing the polar 

 axis until on the boundary itself the stream lines lie along meridians 

 and the twist is altogether on the polar axis. The pitch can be 

 increased up to a certain limit. As this is done the stream lines and 

 the vortex lines fold up towards one another, coincide at a certain 

 pitch, and exchange sides. 



When an external angular pitch of about 330° is attained it is 

 impossible to go further if a simple aggregate is desired. If a higher 

 pitch is desired the aggregate splits into two concentric portions — an 

 inner spherical portion and an onter shell. The central nucleus is 

 Similar to those just described — it produces a part of the required 

 pitch. 



The outer layer has spirals with the same direction of twist which 

 complete the balance of the pitch. In these, however, the motion is 

 in the opposite direction. With increasing pitch this layer becomes 

 thicker and its equatorial axis contracts relatively to the mid point 

 of the shell until another limit is reached ; the stream and vortex lines 

 again fold together, cross, and expand as this second limit is reached. 

 If a larger pitch still is desired there must be a third layer, and so 

 on. The first coincidence of stream and vortex Hues takes place for 

 an aggregate whose pitch is 257° 27'. Whenever a maximum pitch 

 is attained the aggregate is at rest in the fluid. This is first attained 

 when the pitch is 330° 14'. Beyond this there are two equatorial 

 axes. For a pitch 442° 37' the stream and vortex lines again coin- 

 cide, the internal nucleus gives 257° 27' of the pitch, and the outer 

 shell the remainder, and so on. 



At the end of the paper a theory of compound aggregates is 

 developed. It is not worked out in detail in the present communica- 

 tion, but the conditions are determined for dyad compounds, whilst 

 a similar theory holds for triad and higher ones. Each element of a 

 poly-ad may consist of singlets, doublets, &c. The equations of 

 condition allow three quantities arbitrary — as for instance ratio of 

 volumes, ratio of primary cyclic constants, and ratio of secondary 

 cyclic const Kits. The full development of this theory is, however, 

 left for a future communication. 



If we take any particular spherical aggregate with given \ and 

 primary cyclic constant /t, the energy is determinate. We may, 

 however, alter the energy. If it be increased the spherical form 

 begins to open out into a ring form whose shape and properties have 



