Rectilinear Vortices in Ring Formation. 
where 
630 
C =cos(2sr/n); Cl = cos {2(s+4)r/n} ; 
S’ssin {2(s+))a/n}; D= 1—2pC' + p?. 
We now assume a simple solution of the form 
i aerksmijn : Ony = Berksnin ; 
hs41= GB! e2he+3)r in i ey (39) 
and, further, suppose that a, B, &’, 8’ invol 
ve the time asa 
Factor 
Psp axeMs+¥)ni|n 
CEM a AM cM ge (40) 
In simplifying the coefficients we use the following 
summations, valid for =1, 2, ..., n—1, which may be 
proved without difficulty : 
nel a —p”) B 
As n( pi — p-*) 
oO 1—2p0' +p? ie 
iL +p” 2 
“= 2pr(h + p*)C'— 2p} it nt kp*—(n—k)pr-*1 
0 (1=2p0' +p)? 1+p" 
= n®p"( pk — py") 
@ +p”)? 2 
"> 2p(1—p*)S’/E =i 
0 (L~2p0’ $2)? — 1 +p” 
an a) 
ieeapey 1 | er en ohana 
(le a) (41) 
where H = e%(st+)zi/n, 
The 4n equations of the system now reduce to 
Az = PiB+Q'a' + R'B’, 
AB = Pra + R'a!—Q' 8’, 
ne = IPfel! 2 Qa+ RB, 
NB! =Pyla' + Ra— QB, 
(42) 
where 
2n? 
—— cers «' +k(n—k)x, 
; Qn2 prt? 
a 1S 
—— Ff K—h(n—k)pe!, 
(1 sr /a")" ( )P 
2np™ n? py” 
— ie — OA PV 0) oS a ed i] 
«P.= {k(n—h) —2(n Dyes 2) Tesaee 
343 
