The Stability of Fluid Motion. 436 
Equation (7) becomes 
(4-1) y—2 (200k 4 30ty) = (i) (23) 
where k = 5iR/8p'. 
The boundary conditions are 
w=0; dyfda=0; a=tp. 
Solving (23) by a power series }A,«"/n!, we have. 
An+e = 2Ant4—Anse+ 2k (n +1) (n+ 2)(2n+8) Ay, (24) 
As in the simpler cases, it is sufficient to choose fundamental solutions 
involving only even powers of «; denoting these by yo and ye we have 
Wo = 1407/2!+4+a4/4!4+(14+ 12k) a9 /6!4+(14+192h) 28/8! 
+(1+4+1032h) a!/10!+(1+3552k + 20160K?) a! /12!4 (14+ 9492 
+ 696960k?) al4/14 !+ (1+ 21504k + 8162256k?) «18/1614... 
We = 27/214 2a4/4!4 3205/6 !4+(4+4+ 168k) a8/8!+(5 +1656%) «1/10! 
+(6+8184h) o!?/12!+(7 + 28392k + 574560k?) «4/14! 
+ (8+ 78960k + 1120435 2k?) «16/16 !+(9 + 188496% 
+ 10226649647) a!8/18!4.... 
From the boundary condition 
Wody2/da—Wedyro/de = 0, 
we obtain the equation 
p+ 2p?/3!+ 8p? /5 14+ 32p7/7!+128p9/9!4512p"/11! 
+ (2048 + 129024k) p'8/13!+ (8192 +3280896K7) p?/15 ! 
+ (32768 + 7753296k?) p47/17!4+... = 0. (25) 
Using this as an equation for R, we find by trial that the minimum value 
occurs near p? = 3; and the critical minimum value of R is 280 approximately. 
The corresponding value for the ordinary parabolic distribution (m = 1) is 
117. Thus, the critical value of R increases as the flow approximates more 
closely to a uniform stream, without slipping at the walls; and, in this sense, 
the motion becomes increasingly stable. 
8. It has been stated that, under the boundary conditions w= 0, v= 0, 
there is thorough stability, in the ordinary sense, for simple shearing motion 
and probably also for laminar flow between fixed planes. In view of the 
behaviour of actual fluids in similar conditions, another suggestion has been 
put forward by Hopf.* He proposes to express the influence of a wall by 
making the extra normal pressure, due to the disturbance, constant at the 
* L. Hopf, ‘Ann, der Phys.,’ vol. 59, p. 538 (1919), 
174 
