140 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where 
2£=V 2 ^ = spin .... (4) 
using the customary notation. Where the vortex is situated along the axis 
of the cylinder, the motion at any point must be purely a function of r, 
in which case (3) takes the form 
the terms in u and v cancelling out. 
Writing £=Ze~ a * vt and inserting in (5), assuming that Z is a function 
of r only, 
d?Z IdZ 2V n /o\ 
-T~2 + -d-+ a Z = 0 ( 6 ) 
dr A r dr 
This is Bessel’s equation of order zero, the solution which is finite for 
r = 0 being J 0 (ar). 
Hence a solution of (5) is 
£=AJ 0 (ar)e-^K 
If f=0for£ = 0 at r = R, the radius of the cylinder, then J 0 (aR) = 0, 
a transcendental equation determining an infinite series of values of a, Viz. 
^? = -7655, 1-7571, 2-746, . . . 
7 T 
Accordingly we may write for the complete solution of (5) 
£=2 A n' J o Me-W* (7) 
where the coefficients A n are to be found by expanding the initial distribu- 
tion of £ as a series of Bessel functions, as specified above. 
If £ is initially concentrated mostly in the region of r= 0, the first few 
terms only of this series will be of consequence, and these decay on 
account of the term e~ a n vt . Now for water i/ = *01 c.g.s. units, and, taking 
R = 10 cm., say, a = *76 = *24. 
. e — aM __ g-- 006 ^ 
indicating that a relatively considerable time must elapse before £ decays, 
as far as this term is concerned. The terms, of course, decay more rapidly 
as we proceed further up the series, but they themselves become small. 
From the point of view of the present discussion, where we are primarily 
concerned with wjiat occurs in an extremely short interval subsequent to 
a disturbance to the vortex, it clearly suffices to assume that the strength 
of the vortices -f k and — k are sensibly constant.* 
* The full justification for this assumption can ultimately only he found by a com- 
parison with experiment. The experimental investigation of the rate of decay of eddies is 
at present being conducted by the author. 
