ItK 



On the Flow of Elastic Fluids through Orifices. 189 



Substituting this value of V in the preceding couplet and then 

 findinsf the value of rf, we have the following: formula for deferm- 



^ v..v> . W.t^lV^ WA l^j 



ining the densities in the chamber according to the old theory; viz., 



The several densities in the chamber computed by this formula 

 are placed in the fourth column of the table. 



In order to ascertain what the densities in the chamber should 

 nave been according to the new theory, I constructed a formula 

 as follows, preserving the same notation as above. 



By the new theory the force which drives the air through the 



first orifice \^ d — d whenever d is not less than Tj' Butrfis never 



less than - when an equal quantity flows through both orifices, 



for if it were so the chamber would, according to our theory, be 

 receiving as much as could flow into a vacuum under the pres- 

 sure -Jj and must therefore discharge into the receiver as much as 

 "vvould flow into a vacuum under a pressure ^ ; in order to which 

 the density in the chamber must be equal to ^; and therefore 



greater than 5- Consequently, the force which drives the air 



through the ^r5/ orifice is in this arrangement always ^ — d. 

 Again, the force expended in driving the air through the second 



d d 



orifice by the new theory is 5 whenever D is not greater than 5- 



Let us first construct a formula for the cases in vrhich D is not 

 greater than -• In these cases the densities under which the air 



passes the orifices are respectively J -d and 5. Since the forces 



are as the velocities, ^-d l ^i'.'.Y : v, and since the quantities are 



2 



dv 



equal, dY=—, and V=|- Substituting this value of V in the 

 couplet, we have d=-J', a constant quantity. 



Hence 



density in the receiver varies from to ^^, the density in the 



5 



chamb 



4 



-^. Let us now 



5 



construct a formula for finding the value of d when D is greater 



