PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS, 
769 
Since 
P~ 
k^+kj* 
k**-k**— 
1 + x n 
1 —x n 
Further denote l+x+ . . . + x n ~ l by X* and (1— af) + (l + x n )p by f,. It is to be 
noticed that since x lies between 0 and 1, these quantities are both positive and finite. 
With this notation 
and is always negative and finite. 
The second coefficient is 
A 
l +jpp 
{np(n -{- 1 )iv 2 p-\-n(n —p)v 2 ffi nh.apv-\- A?d 2 
After some algebraic reduction this may be put in the form 
/«L 
(1 -\-x v )n{n -\-1 )urp +(1 — x)v 2 t 0 n ~ 
n- 
2r + 3 
-j-r 2 +3r+ 
Here c is always finite and negative. 
The third coefficient is 
d-- 
nAa 
1+PP 
2 nX H 
{(?i -f 1) w-p -f -(np — 1) v 3 + A av] 
f n _ 
(„+l )(^-v*) + 
2 n 
which again, after some algebraic reduction, becomes 
d= 2 f £ *{(»+1 )(»V -o' 1 ) +S 7\n-r- 1 yj v 
This is always finite, and if iv 2 p>v 2 it is also always positive. But if w~p<v'~ it 
becomes negative for values of n less than a particular value, depending on the value 
of vf-'p/v 2 . 
The conditions that 
X 3 -f- bX 2 -\- cX-f- c?= 0 
shall have real roots are 
i. 
ii. &V-4c 3 > (46 s —186c+ 27d)d 
We have already seen that c is always negative. Hence the first condition is 
satisfied under all circumstances. 
