152 Lord Rayleigh on the Instability of a Cylinder 



in which, however, we may effect further simplifications by 

 means of the properties of J . We find by use of (26) 



J --W'-W =Jo2+J , 2 ( 1+ _i_), 



so that, finally, 



._ T(l-ffqg) ,« n 



m 2/ia.Fa 2 {J 2/J '2 + l + l/Fa 2 } , "■" {0L > 



In (31) the argument of J , J ' is ika, or z as we will call it 

 for brevity. And by a known property J '= — Ji< Now 



J()W = 1-32 + O 



22 ■ 2 2 .4 2 ' 



J iW = l V"2T4 + 27i^~6 ""•'•} ; 

 so that if x = ka 



x 2 # 4 



J (ix) =1 + ^ + 22712 - • • '9 



Ji (w) = f I 1 + ^4 + OT6 + • * * } ' 



These functions have been tabulated by Prof. A. Lodge* 

 under the notation l Q (x), Ii(x), where 



Io W = Jo0>) = 1 + fa + 22^42- + (32) 



IiW=-^Ji(^)=||_ 1 +2^i+ 2.1*. 6 * "*J ' * ^ 

 In this notation 

 x^J *{ix)IJ 1 *(ix) + l + l/x*} = x*+l-xn o %v)/I 1 %x), . (34) 



and we have to consider the march of (34) as a function of x. 

 When x is very small, 



I (*) = l-i.* 2 , Ux) = ix + T \x% 

 so that 



(34) = — 3 + terms in x 4 ; 



and then from (31) 



in =w« (35) 



* Brit. Ass. Report, 1889, p. 28. 



