PROFESSOR W. M. HICKS OK YORTEX MOTION. 
49 
If the volumes of all the layers are equal, 
Hence 
Bp = - i (P — 1) —ip — I)"'"'} . 
or 
p (p - 1) [p'' - ip - If'l = {{p - + .... + Xi}. 
Now 
Xp-i — s' {^’p-i ^p)- 
Hence 
{{p-l)p^>^ 
-{p + K= -1(p -ifVi- 2;(p-2fV2+ •... + x,] 
^ - I [((P - !)•'" - (p - 2)“'*) i,-. + ■•••+ (2“’ - l)fc + k]. 
or subtracting two consecutive equations 
ir' - (p +1) (p - in h =-{(p- 2)“ - (p -1) (p - in 4- 
Thus the k can be determined in order from the inside. The peculiarity is that the 
process can stop at any point. That is that if we have two poly-ads, with m and n 
layers respectively (w > n) then the first n layers in the first will be precisely similar 
to those in the second. The values are 
and when p is large 
h, — — ^ ^ ^k^ — — l-3120^’i 
k^ = -f 1*4717^1 
ki = — l’5866^’i 
kp — ■ kp_i. 
As another example, take the case where the layers are formed of the same 
material, fie., the vorticities alternately equal and opposite. Then k^ = ( — 
Xp = f fip = I fi'i { — Y~' but X,, = i Xq = (—i fi’i 
- «i + .... + = i 
C(l - o^_i 
Let denote the ratio dp+i/a-iy. 
These values are then given by 
nP ^ _1 —4)1 
^ +::P 
K-i 
_ 2 
and may be found in succession. The equations are, if denote 1 
®p-i 
+ 
VOL. CXCII.—A. 
a^p (a^p + 1) = f 6p (4 + aJp + !)■ 
H 
