1907-8.] On the Theory of the Leaking Microbarograph 
441 
Then, by Poiseuille’s Law 
dv 
,R 4 
<«=8^-% = A W-^ 
(1), 
where 
A - 7rR 4 /8^L . 
We suppose the temperature kept constant. Hence, since the leak is 
very slow, we have 
(p + dp)/p = (v + IJ -±^-dvyV; 
that is, 
'P 
do _ 2 V dp 
dt p + EFT dt 
( 2 )- 
Hence the equation which determines the variation of the pressure is 
that is, 
2 V dp _ 
p + Z3 dt 
A(ST - p) ; 
dp = \jp+^) ) 
eft 2V v ' 
(3). 
Now, in the case of the microbarograph, tef —p is always small. Hence 
we may for our purposes replace^) and tef in the factor p + TEF by 2tef 0 . The 
equation (3) then becomes 
S = ..... (3 
or, if 
1//a= r = V/ATETq = S^LV/TrEEfgR 4 .... 
dp 
•\ • * 
) 5 
dt 
+ pp — ptn 
( 4 ) . 
(5) , 
the solution of which is 
P = (A + /jl dtzsef^e ~ ^ . 
( 6 ), 
where A is an arbitrary constant. 
The following special cases are of practical importance : — 
Case 1. Suppose air blown into the chamber until the pressure becomes 
p 0 , and this is allowed to leak out again until the barometric pressure falls 
to xSq. Then, since p=p 0 when t = 0, and tef = tef 0 throughout, we have 
rt 
PMPo + ^0 
dtet xt )e~^ t 
= {Po + ^ r o( e ^- 1 )}e-^ 
= ^0 “t (Po ~ ^o) e ^ 
(7). 
* The exact equation (3) can be reduced to the form +y 2 =f(t), and could be used 
instead of the approximate one ; but the calculations are much more complicated. 
