48 
PROFESSOR W. M. HICKS OH VORTEX MOTION. 
8. Polyads .—Passing on now to the consideration of any number of layers, let the 
radii of the spherical boundaries from the inside outwards be denoted by Uj, cto. . . ; 
the vorticities by k^, and the stream-functions by xjji, • V'/t \p. Then 
xpi — 27r {AiP ~ -g- } sin^ 6 '] 
^ — 1^/’^ jsiiP 6 ^ 
B 
xjj,,+i = 27t silP 6 
J 
Applying the conditions of continuity at the 2 ^th boundary, there results 
■^2,^1 J' — \ — Ap+ittp -f- 
a„ 
1 k 
5 ) 
■P 
B„ 
with 
Adding 
Similarly 
2AX - ^ - I=■ 2A,,^xa| - ^ 
Up Up 
5 
Bj — 0, A,j+i — 0. 
A^,+i ~ A^ = (4,+i — t^) ol with A,,+i = 0. 
B^,+i — = — ^^5- {k^^2 — kp) al with Bi = 0. 
Clearly the A’s evolve from the outside, the B’s from inside. 
Write 
F (^iJ “ ^p+i) = 
Then 
A^, — kp^2 — with A.^+1 = 0, 
•B^,+i — B^, = I \pal with Bi = 0. 
Hence 
Ap = t;\pal, Bp = ^tr%(i^ 
Thus the \p are completely determined. 
For steadiness of motion it is necessary that the translatory velocity of the different 
boundaries be the same. This is obtained if the velocities of the inner and outer 
boundaries of each layer are equal. 
Hence we get n — 1 equations (p = 2 to 11 ) 
iV 
kp A J- — ^ kpCCp — kp + 
i I rr- 
'i)-i 
or 
or 
B 
-:K) 
5 
B,= 
Up U,p_ 
