﻿[ 67 ] 

 XI. Proceedings of Learned Societies, 



ROYAL SOCIETY. 



[Continued from vol. xl. p. 460.] 



June 16, 1870. — General Sir Edward Sabine, K.C.B., President, in 



the Chair. 

 HpHE following communication was read: — 



1 " On the Mathematical Theory of Combined Streams/' Bv W. 

 J. Macquorn Rankine, C.E., LL.D., F.R.SS. Lond. and Edinb. * 



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

 combined streams were to a certain extent investigated by Venturi, 

 and stated in his essay ' Sur la Communication laterale du Mouve- 

 ment dans les Fluides' (Paris, 1/98). The principle of the conser- 

 vation 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 contained in 

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

 1863). The theoretical conclusions are tested by comparison with 

 experiment, and applied to practical questions, especially those rela- 

 ting 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 principle, being effected at one ope- 

 ration by the direct application of the principle of the equality of 

 impulse and momentum, instead of by the consideration 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 

 component 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 cham- 

 ber, 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 discharge-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 



F2 



