90 



Dr. W. J. M. Rankine on the 



[Recess, 



" On the Mathematical Theory of Combined Streams." By W. J. 

 Macquorn Rankine, C.E.j LL.D., F.R.SS. Lond. and Edin. 

 Received Sept. 10, 1870. 



1. Object of this Investigation. — The principles of the action of com- 

 bined streams were to a certain extent investigated by Venturi, and stated 

 in his essay 'Sur la Communication laterale dn Mouvement dans les 

 Fluides' (Paris, 1798). The principle of the conservation of momentum, 

 so far as I know, was first explicitly applied to combined streams by Mr. 

 "William Froude, F.R.S., in a paper on Giffard's Injector, read to the 

 British Association at Oxford, in 1860, and published in the Transactions 

 of the Sections, p. 211. Various other authors have treated the same 

 problem by different methods, based virtually on the same principle. 

 A very complete and precise investigation of the theory of combined 

 streams, in every case in which two streams only are combined, is con- 

 tained in Professor Zeuner's treatise ' Das Locomotivenblasrohr " (Zurich, 

 1863), The theoretical conclusions are tested by comparison with experi- 

 ment, and applied to practical questions, especially those relating to the 

 apparatus from which the treatise takes its name. The object of the present 

 investigation is to apply similar principles to the combination of any 

 number of streams ; and the demonstration of the fundamental dynamic 

 equation differs from that given by Zeuner in method, though not in prin- 

 ciple, being effected at one operation by the direct application of the prin- 

 ciple of the equality of impulse and momentum, instead of by the con- 

 sideration of the loss of energy that takes place during the combination 

 of the streams. 



2. Terms and Notation used, and Suppositions made. — The several 

 streams which are combined will be called before their junction, the com- 

 ponent streams ; the stream formed by their combination will be called 

 the resultant stream. The passages through which the component and 

 resultant streams flow will be called respectively the supply-tubes and the 

 discharge-tube. The combination of the streams will be supposed to take 

 place in a short cylindrical chamber, with its axis parallel to the direction 

 of flow, which will be called the junction-chamber. 



At one end of the junction-chamber are the outlets of the supply-tubes, 

 which will be called the nozzles ; at the other end, the inlet of the dis- 

 charge-tube, which will be called the throat. It will be supposed, further, 

 that the supply-tubes are so formed as to direct the component streams at 

 the nozzles, so that they shall all flow sensibly parallel to each other and 

 to the resultant stream. The principal symbols used are as follows : for 

 any one of the component streams : — 



a, area of nozzle ; 



v, velocity of flow at nozzle ; 



* , bulkiness, or reciprocal of density at nozzle. 

 The several component streams may be distinguished from each other, 

 when required, by suffixes ; as 1, 2, 3, &c. 



