454 
Proceedings of the Royal Society 
where X = ^ U .... (51), 
Jcd 
and q 2 = m 2 a 2 ^ "A - . . . (52). 
Kemarking that J 4 (g) is the same for positive and negative values of 
g, and that it passes from positive through zero to a finite negative 
maximum, thence through zero to a finite positive maximum and so 
on an infinite number of times, while q is increased from 0 to oo , 
we see that while X is increased from - 1 to 0 the first member of 
(50) passes an infinite number of times continuously through all 
real values from — oo to + oo : and that it does the same when X is 
diminished from + 1 to 0. Hence (50), regarded as a transcendental 
equation in A., has an infinite number of roots between — 1 and 0 
and an infinite number between 0 and + 1. And it has no roots 
except between — 1 and + 1, because its second member is clearly 
positive, whatever be ma ; and its first member is essentially real 
and negative for all real values of X except between - 1 and + 1 , 
as we see by remarking that when A. 2 >1, — q 2 is real and positive, 
and - is real and >i/( - q 2 ), while i/q 2 X, whether 
positive or negative, is of less absolute value than i/( - q 2 ). 
Each of the infinite number of values of X yielded by (50) gives, 
by (51) and (13), a solution of the problem of finding simple har- 
monic vibrations of a columnar vortex, with m of any assumed 
value. All possible simple harmonic vibrations are thus found : and 
summation after the manner of Eourier for different values of m, 
with different amplitudes and epochs and different epochs, gives 
every possible motion, deviating infinitely little from the undisturbed 
motion in circular orbits. 
The simplest Sub-case, that of i = 0, is curiously interesting. Eor 
it (50), (51), (52) give 
and 
J 0 (g)_ - £o( ma ) 
gJ 0 (g) mal 0 {ma) 
2(j)ma 
1 J(m 2 a 2 + q 2 ) 
. (53), 
. (54). 
The successive roots of (53), regarded as a transcendental equation in 
q , lie between the 1st, 3d, 5th - - - roots of J 0 (^) = 0, in order of 
ascending values of q, and the next greater roots of J' 0 (g) = 0, 
