1870.] 



Mathematical Theory of Comlined Streams, 



91 



For the resultant stream : 



A, area of throat ; 



Y, velocity of flow at throat ; 



S , bulkiness, or reciprocal of density at throat. 

 Intensities of pressure, in absolute units on the unit of area : 



p , at the nozzle end of junction-chamber ; 



P , at the throat. 



(These may be converted into units of eight on the unit of area, by 

 dividing by g). 



The flow of each stream is supposed to be steady. The fluids may be 

 either liquid, vaporous, gaseous, or mixed. 



3. Equation of Continuity. — The mass of fluid that enters the junction- 

 chamber through a given nozzle in a unit of time is — • The mass 



. AV 



discharged in the same time at the throat is . The flow being steady, 

 the following equation must at every instant be fulfilled : 



0) 



If S and the several values of s are given, that equation gives the velo- 

 city of the resultant stream in terms of those of the component streams ; 



Viz ' V=A.25.™ (I A) 



A * 



If all the fluids are liquids, each of sensibly invariable bulkiness, we 

 have also AV=S . av, that is, the volume of flow of the resultant stream 

 is equal to the aggregate of the volumes of flow of the component streams ; 

 but if any or all of the streams are vaporous or gaseous, the values of * 

 will depend upon that of p , and the value of S upon that of P , and upon 

 the changes of bulkiness of the fluids which may take place in the junc- 

 tion-chamber, through change of temperature, change of condition, or 

 chemical action. 



In any case S may be regarded as a given function of P , and of the 



av 



mutual proportions of the several values of — ; in other words, of the 



ingredients in the resultant stream. 



4. Dynamical Equation. — The aggregate momentum of the mass of 

 fluid that enters the junction-chamber through the nozzles in a unit of 

 • • av" 



time is 2 . — . The momentum of the equal mass which leaves the junc- 



• AV 2 



tion-chamber through the throat in the same time is -« — 



The forward impulse exerted in a unit of time upon the mass of fluid 

 in the junction-chamber by the pressure at the nozzle end of the chamber 

 is^ A. The backward impulse exerted in the same time on the same 

 mass by the pressure at the throat-end of the chamber is P A. By the 



