PROFESSOR W. M. HICKS OR VORTEX MOTIOR. 
95 
Hence, when Xj is large, the outer layers have their pitches at the axes about 
7rv/2 = 254° 31'. 
Fig. 5, Plate 2, shows the relative positions of the shells and axes for the X? and Xi 
aggregates. The thin lines belong to the Xo, the dotted to the Xj. A, A are the 
position of the X., equatorial axes. B, B those of the Xj. 
27. In the preceding investigation we find doublets, triplets, &c., naturally arising. 
We may have also built-up systems consisting of monads, dyads, &c., as in the cases 
developed in the previous section. Each element of a poly-ad may coiisist again of 
singlets, doublets, &c. I do not propose now to develop this theory of multiple 
combiDation to any length, but merely to draw attention to it, and to determine the 
necessary conditions for the case of a dyad only. 
Beferring to § 15, the general solution of the differential equation contains not only 
J functions, but also the functions Yg = + sin y, which are suitable only for space 
not containing the origin. They are therefore suitable for any shell embracing an 
interior aggregate. In the shell the functions will be of the form AJ -p BY, or as it 
may be written 
sin {a + y) / , ^ 
-cos (a -p y). 
It will be convenient to denote this by f (a, y). 
Let now the radius of the interior aggregate be a, that of the exterior h. 
also X, X' denote the corresponding parameters. 
Then we may write 
— “ Jx| sin^ 6 . 
Inside = L IJ 
Let 
(39), 
Shell ^\so = qL\f[a,~)—— /(a, X') \ sin“ 9 
• (M). 
Outside xfj = — 77 Y (— —■) sin“ 9. 
At the interface i//, = i />2 and if/i = 0, therefore 
b / ¥ 
Write alh=p. This equation, when developed, gives 
J(X» -/J(V) 
tan a = — 
Y(Vj9)-/Y(V) 
( 41 ). 
Moreover, the tangential velocities must be the same. Hence, Avhen r — a, 
dxl/Jdr = 
