435 Prof. T. H. Havelock. 
when « =p. The series for 3 is given in §5; also we have 
tha = 07/214 2a4/4!4+(3 45h) 08/614 (44 28h) 08/8 | 
+(5 + 90K + 65K?) 19/10 !+4 (6 + 220k + 606K) «12/12! 
+(7+ 455k + 3037k? + 136543) «4/14 !4(8 + 840+ 1096822 
+17880K) a!6/16!+.... 
The boundary equation (18) leads to 
2/4 1+ 8p2/6 !-432p'/8!+ 128p8/10 + (512 + 28024) p8/12! 
+ (2048 + 3136?) p/14!+ (8192 4+ 252162) p2/16 ! 
+ (32768 + 1740802?) p'4/18!4+...= 0. (19) 
The minimum value of R seems to occur for about p? = 12, though it is 
not a sharply defined minimum ; however, with a similar approximation as 
in previous cases, we find the critical minimum of R to be 110. 
7. It is well known that, when fluid motion through a tube has changed 
from laminar to turbulent flow, the distribution of mean velocity over the 
cross-section alters so that the velocity becomes more nearly uniform over 
the greater part of the section, while falling to zero at the walls. This 
suggests a study of the comparative stability when the distribution of 
velocity alters in this manner, the boundary conditions being unchanged. 
However, it must be noted that we assume the distribution to be a steady 
state which has been acquired under a. law of force, which may be deter- 
mined from the hydrodynamical equations, so as to give the required form 
for U. 
A simple form, which illustrates the points in question, is 
U = (141/2n)(1—7), (20) 
As 7 is made larger, the velocity approximates more closely to the mean 
velocity, U, over the greater part of the cross-section, while remaining zero at 
the walls. The corresponding law of force is, in the usual notation, 
X = v(4n?—1)(U/a?) n*-2, (21) 
The greater the value of n, the more is the field of force concentrated near 
the walls, quite apart from the value of the viscosity. The flow approxi- 
mates to a uniform stream, but retaining the condition of zero velocity at 
the walls. 
The usual case of flow under a uniform field of force is given by n = 1. 
It is sufficient for comparison to work out another numerical case, say n = 2. 
We have then 
U = $(1—n'). (22) 
173 
