in certain Problems of Viscous Fluid Motion. 625 
Substituting in (7), we have 
Tey cab +k(1 +h) 
t 2 
ON _ Aug > ER perma a ae 
o 
e 7 vARt/h2 
SS 2 
g NM +KL+%)’ ey) 
the terms independent of ¢ cancelling out on summation. 
The velocity at any time is given by 
_ Agph 1—e-rrt/re 
ay nan = OD 
It can be verified by summation that the limiting steady 
velocity has the value goh/2u. The fluid velocity at any 
point can be obtained by substituting V from (21) in (2) 
and reducing the expressions, but it is, of course, simpler to 
insert suitable functions of x directly in (21) ; we obtain 
aes gah (1 =) _ Agph? & sin {AC —a/h)} e- rene (22) 
ES UY yer rears. 7 
In this particular problem the result ean also be obtained 
from the differential equation together with the boundary 
and initial conditions, by assuming the existence of a 
limiting steady state. In the preceding analysis the 
existence of a final steady state is associated with the occur- 
rence of zero as one of the exponents in the nucleus of the 
integral equation (4). 
5. Itis interesting to deduce the motion in an infinite 
fluid from these results. In solving this case directly, 
Rayleigh obtains the equation of motion as 
dV in Zpv? (* V'(r) dr & 
dt om), /(t—7) 7° 
(23) 
Applying Abel’s theorem, this is reduced to an ordinary 
differential equation whose solution is given as 
Am upV | 90 =4pviti—n0 + Zo ewe | Oa, (EY) 
< 2pv8t?/o 
We obtain (23) from (4) by giving the nucleus its limiting 
value, since’ 
Lim (2u/oh) & e-n nv (t—2)/n2 (2pv?/om) ( e-2%(t-=1) dy, 
h>w —o oJ -a 
