1870.] Mathematical Theory of Combined Streams. 93 

 and the expression for the loss of energy becomes 



s MP (3C) 



When the fluids are all liquids, whose compressibility may be neglected, 

 we have I P ° S^P = S (P — p ) ; and substituting for the difference of 



JPo 



pressures its value, according to equation (2), the following expression is 

 found for the loss of energy at the junction, 



that is to say, in the case of liquids all the energy due to the several velo- 

 cities (»— V) of the component streams relatively to the resultant stream 

 is lost. 



When the expression (3 D) is reduced to a single term, it becomes the 

 well-known value of the loss of energy of a single stream of liquid at a 

 sudden enlargement in a tube. 



6. Efficiency of Combined Streams. — The efficiency of a set of combined 

 streams may be defined as the fraction expressing the ratio borne by the 

 total energy of the resultant stream after the combination to the aggregate 

 energy of the component streams before the combination. It is expressed 

 as follows :- 



>P 1 



(4) 



ay r v 2 C J 



So t 8 Jo 



m+t **)>' 



7. General Problem of Combined Streams. — In most cases the problem 

 of combined streams takes one or other of the two following forms. In each 

 of the two forms the areas of the nozzles a v a 2 , &c. are given, and also the 

 area of the throat, A. 



First Form. — The quantities given, besides the before-mentioned areas, 

 are the pressure at the nozzles, p of and the velocities of the component 

 streams, v v &c. The functional values given are those of s Q , v s , 2 , &c, in 



terms of p Qi and of S in terms of P , &c. Those functional 



*0» 1 S 0> 2 



values are to be substituted in the equations (1) and (2) ; and the solution 

 of these equations will give the numerical values of Y and of P . In the 

 case of liquids of sensibly constant bulkiness, s , 1 &c, and S are quan- 

 tities sensibly independent of p and P ; and then equations (1) and (2) 

 can be separately solved without elimination, giving respectively Y and P . 



Second Form. — Each of the component streams flows through a passage 

 whose factor of resistance, /, is given, from a separate reservoir in which 

 the pressure p and the elevation z of the surface above the junction- 

 chamber are given. The resultant stream flows through a passage whose 



VOL. XIX, 



