625 Prof. T. H. Havelock on the Stability of 
where 
- ps nk(p®—p-*) n2gn—k( 1—p*)? 
P=k(n—k) ee -4 SS. 
kp — pret) npr" +p*)? 
=k(n—k)+2(n+1 ane + aa 
Q= kn) +2(0 41) + MOREE won 
k na —k 2 ,>n—k Spek 
[ec oe ar) 9) gs 
T=)" (=p? 
We may check these expressions by deducing the equations 
for the corresponding disturbance of an infinite double 
symmetrical row of vortices. If d is the distance between 
consecutive vortices in each row, and h the distance between 
the two rows, we have, in the limit, 
2najn=d; 2kn/n=¢d; 
p=(1+ 27h/nd)-}. 
With these (17) gives the limiting value of the linear 
velocity of the vortices, namely, 
K wh 
5d coth ape 
further, the quantities n2P/272, n?Q/272, and n2R/27? become 
respectively the quantities A+C, A—C, and B in the 
notation of Lamb’s ‘ Hydrodynamics,’ (5th ed. p. 221). 
Returning to equations (24), we take « and 8 proportional 
err /4ra® 
and obtain 
A=-—7R+(PQ)?. . Teeny, (26) 
For complete stability the product PQ must be negative, 
or zero possibly, for all the values of k. To prove instability 
it is sufficient to show that PQ is positive for one value at 
least of k&. From the form of the expressions in (25) we see 
that P and Q are symmetrical in & and n—A, and that the 
critical mode to examine is k=4n for n even, or k=}(n+1) 
for n odd. 
For n even we have 
P(dn)=3n2—n2pi/(L1+p)2, . . . (2%) 
which is always positive. Further, 
4n n2p” 
+ =>, + (28 
1 yp ( iL — p)* ( ) 
Q(3n)=4n2 + 2(n +1) — 
338 
