Rectilinear Vortices in Ring Formation. 624 
we obtain eventually the equations 
Ama? ry ale 2n2p™ . ho 
Ka BGPP 7 tps 
ed ead eS hg 
ost LI=C 7 (1 = 290 +p? Pie 
—"5 2pA=p)8 rea 
s=1 (lL—2pC +p?)? a 
Ana; ¢n?—1 2n*p” 
A= { a geoy 
An =O yet) } rs 
lap =a 
& 'S 4 T_ 2p{(l + p?)C— 2p} ). resi 
s=1 (1—C (1—2pC + p?)? a 
— "3 _2p—p*)8 
s=1 (1— 2pC-+ p*)? 
There are 2n equations of this type, and we examine now 
a possible solution of the form 
1541/4 wer**smiln ; 0,41 = Berni/n, LR INCeNRy ate (22) 
Wat PD), Ty 5 PIL, 
On substituting these expressions we obtain two equations 
ina and B. In simplifying the various coefficients we use 
the following summations, whose proof need not be given 
here :— 
= (1—p?) cos (2ks7/n) ne n(p + p-*) anak 
a L—2pC +p? ye 1l—p 
'S (1—pC) cos (2ks/n) = n(p* +p"-#) ieee 
1 1—2pC +p? 2(.1—p") 1l—p 
= {(1+ p?)C—2p}cos (2ks7r/n) _ nk (pk-1 — pn-#—-2) 
Bee T Oe eal hy Ba a(O 1) 
1 (1—2pC0 +p)? 2(1—p”) 
n®p”—(p*—p-*) 1 
+ _ ans vmstat P33 
BUS prt siCleep)tsVunhi a 
valid for 0 <p < 1, and k=1, Py acy lly 
We obtain after some reduction the equations 
(47ra*/«)«e=P@—iRe, 
(41ra*/x)B = Qa—iRZ, 5 3 6 (CX) 
