52 PROFESSOR W. M. HICKS OK VORTEX HOTIOX. 
The distance from the inner layer is therefore 
From the outer it is 
But (p. 50) 
therefore 
ftp 1 — ^^p-\ 1 
Ratio = Pf-.r 
•'-S-l ■ ^ 5 Jp-V 
^p-\ = f + 
Ratio = 1, 
or the equatorial axis, with increasing number of layers, tends to bisect the distance 
between the two boundaries of the layer. 
10. Energy .—The energy within any region is 
the integral extending within the boundary of the region. By the ordinary method 
this is reduced to the form 
E = - ^ c/s + ff wi/zr/p dz. 
Ttt J p cm J J ' 
Since 
cl /I 
clp p dp 
. V7 + 37 
d / I c/yx 
dz 
dz 
Tttco. 
If the boundary be infinite and the fluid at rest then the first integral is zero, 
and 
If the 
E = j j anjj dp dz. 
The integral extending only to spaces which contain rotational motion, 
motion IS of uniform vorticity co = I'p, and 
E = ^ II pi// c/p c/z. 
In the cases here considered xJj is of the form /(r) sin" 0, and 
E = 2k 11^ E'f^r) dr sin® 6 dO =■ ^k^ 
In the case of a poly-ad f{r) is different for the various layers, and 
E = t j/^’i 1^' r-f, (r) dr + h | Efo. {r) dr + . . 
