Q6 Lord Rayleigh on the Question of 



full-blown exponential sort, is thus excluded, provided d 2 W/dx 2 

 is of one sign throughout the entire region of flow limited by 

 the two parallel plane walls. 



Commenting upon this argument, Lord Kelvin * remarks 

 that the disturbing infinity, which arises in (13) when n has 

 a value such that n + TcW vanishes at some point in the field 

 of motion, " vitiates the seeming proof of stability." Perhaps 

 I went too far in asserting that the motion was thoroughly 

 stable ; but it is to be observed that if n be complex, there is 

 no " disturbing infinity.'"' The argument, therefore, does not 

 fail, regarded as one for excluding complex values of n. What 

 happens when n has a real value such that n + JcW vanishes 

 at an interior point, is a subject for further examination. 



The condition for two dimensions that dPN /dx 2 is of one 

 sign throughout is satisfied for a law of flow such as that of a 

 viscous fluid, and we shall see that the corresponding condi- 

 tion for (17) in the more general problem is also satisfied in 

 the case of the steady flow of a viscous fluid between cylin- 

 drical walls at i\ and r 2 . The most general form of W for 

 steady motion symmetrical about the axis is t 



W=Ar 2 + Blog?> + C, .... (19) 



in which the constants A, B, C are related by the conditions 



0=A?y> + Blog^ + C, 

 = Ar 2 2 + Blogr 2 + C. 



From the last two equations we derive 



A(r s «-r 1 i )+Blog»- !! /r 1 =0, . . . (20) 



so that A and B have opposite signs. Introducing the value 

 of W from (18), we obtain as the special form here applicable 



, 4s 2 A-2ft 2 B 

 #V 2 + s 2 ' 



which is thus of one sign throughout the range. A small 

 disturbance from the steady motion expressed by (19) is 

 therefore not exponentially unstable. 



The result now obtained is applicable however small may 

 be the inner radius i\ of the annular channel. But the exten- 

 sion to the case of the ordinary pipe of unobstructed circular 

 section may be thought precarious, when it is remembered 

 that provision must be made for a possible finite value of u 

 when ^ = 0. But although a and /3 may be finite at the lower 



* Phil. Mag. Aug. 1887, p. 275. 

 t Basset's ' Hydrodynamics,' § 514. 



