the Stability of the Flow of Fluids. 65 



We will now consider (11) in the abbreviated form, 



where a is a positive number not less than unity ; or, again, 

 <Tr{? dr) +br U = ^+kW • • (i3) 



The question proposed for consideration is whether (13) 

 admits of a solution with a complex value of n, subject to the 

 conditions that for two values of r, say r x and r i} it shall 

 vanish. This represents the flow of fluid through a channel 

 bounded by two coaxal cylinders. 



Suppose, then, that n is of the form p + iq, and u of the 

 form ol + ift, where p, q, a, /3 are real. Separating the real 

 and imaginary parts in (13), we get 



^) + ^- 2 «= b+wf + f {(P + *W)« + ^}, (14) 



dr \ dry 



and thence 



B d( da\ d( d/3\_ kr«W 1 (« 2 + /3 2 ).q ( 6) 



We now integrate this equation with respect to r over the 

 space between the walls, viz., from r x to r 2 . The integral of 

 the left-hand member is 



***--£ w 



and this vanishes at both limits, 13 and a being there zero. 

 The integral of the right-hand member of (16) is accordingly 

 zero, from which it follows that if Wi be of one sign through- 

 out, q must vanish — that is to say, no complex value of n is 

 admissible. 



The general value of W l9 viz., 



rf 2 W__l dW tfr 2 -** 



dr 2 r dr k 2 r 2 + s 2 > * ' ' * "*' 



reduces in the case of two dimensions to d 2 W/dr 2 , or, as we 

 may then write it, d 2 W/dx 2 . Instability, at any rate of the 

 Phil. Mag. S. 5. Vol. 34. No. 206. July 1892. F 



