Vibrations of a Columnar Vortex. 165 



The condition that the liquid extends to infinity all round 

 makes w=Q when r=co . Hence the proper integral of (24) 

 is of the form & : and the condition of undisturbed continuity 

 through the axis shows that the proper integral of (13) is of 

 the form J,. Hence 



w = CJ/vr) for r < a . 1 , q j 



tK '}.,...„. (48) 



and w=€$ i (mr) „ r>a } ) 



by which (47) becomes 



Cuo—n) (uo—n) / y + */ / \ 



v y L v Ji(va) a J _ — mWi(ma) . 



4<» 2 -(ift)-w) 2 "" if. (ma) ; ^ 



or by (15), 



where 



and 





2« 



t(P — 71 



X -"97T^ • ( 51 ) 



1-A 2 

 X 2- 



q 2 = m 2 a 2 — 2 —. ...... (52 



Remarking that J z (^) is the same for positive and negative 

 values of q, and that it passes from positive through zero to a 

 finite negative maximum, thence through zero to a finite po- 

 sitive maximum, and so on an infinite number of times, while q 

 is increased from to go y 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 

 + co , and that it does the same when X is diminished from 

 + 1 to 0. Hence (50), regarded as a transcendental equation 

 in X, has an infinite number of roots between — 1 and and an 

 infinite number between and + 1. And it has no roots except 

 between —1 and + 1, because its second member is clearly po- 

 sitive, 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 X 2 >1 — q' 2 is real 

 and positive, and —J f i(q)/qJi(q) 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 harmonic vibrations of a columnar vortex, with m of 

 any assumed value. All possible simple harmonic vibrations 



