' E.W. Blake on the flow of Elastic Fluids through Orifices. 81 
fore when VD is a maximum 4/( AD—D?*) must likewise be a 
maximum. Now when (AD—D2?) is a maximum Das 
Hence when the discharge is into a vacuum, the density of the 
fluid at the orifice is equal to half the density in the discharging 
vessel. 
For convenience in the illustration, we have made the passage 
from one vessel to the other in the figure, divergent each way 
from the orifice; but our reasoning would obviously be equally 
applicable if the orifice opened directly from one vessel into the 
other, without the intervention of the divergent tubes. 
Let us now inquire what will be the value of D when the 
, Teceiving vessel contains fluid of any density less than that in 
a the discharging vessel. 
= _-Let the density in the receiving vessel be d. Then d isa 
limit beyond which the fluid cannot expand, either before or after 
‘It passes the orifice; so that D can never be less than d. As in 
the preceding case “(AD -D?) was the maximum for all the 
values of D that can be assigned from cypher to A, s0 in this 
case, and for the same reasons, ,/(D— D7”) must be the maxi- 
mum for all the values of D that can be assigned between d and 
AQ. But we found in the other case that the maximum occurs 
when =}. If then this value of D is assignable between d 
ah i 
: A 
and A, the maximum must in this case also occur when D= 2° 
But this value of D will always be assignable between d and A, 
if d be not greater than >. Therefore if d be any quantity not 
Steater than = D will be equal to S. In other words, if the 
€nsity in the receiving vessel be not greater than half the den- 
sity in the discharging vessel, the density in the orifice will be 
equal to half the density in the discharging vessel. = 
_, Again, from the nature of maxima and minima, it is obvious 
that “(AD- D2) will be a maximum when, of all the values 
that are assignable to D, that value is assigned which differs least 
.: a A . 
from 9° Hence, if dexceed 3 so that D must have a value 
re iter than 2. then ./( AD — D?) will be a maximum when D 
has the smallest value that is assignable to it. Now the smallest 
-Yalue that is assignable to it in this case is D=d. Hence, if the 
Nsity in the receiving vessel exceed half the density in the dis- 
ging vessel, the density under which the fluid passes the 
@ is equal to the density in the receiving vessel. 
ae 
_ Seconp Serres, Vol. V, No. 13.—Jan., 1848. 
