256 REPORT — 1876. 



Prom this last by (4) and (3) we get 



nhi,—nk = h', or n(hi—k)=h' ; 



whence by (5) h' = nh (7) 



From this, if we put P and P' to denote total pressures on homologous small 

 areas at U and U', it foUows that 



P'=»T (8) 



This holds good for any homologous places in any homologous guide-tubes, 

 and so it holds for immediately adjacent places in any two contiguous guide- 

 tubes. Hence, in respect to any small element of the septum between two 

 adjacent guided stream-filaments in Flow No. 1, considered comparatively 

 with a homologous element of a septum in Flow No. 2, the homologous dif- 

 ferential pressures in No. 1 and No. 2 wiU be as 1 to n^ 



Thus the demonstration is now completed of aU that is included in Pro- 

 position A ; and we are ready to go forward to the demonstration of Theorem 

 II. for which Proposition A was meant to be preparative. For this we have 

 to observe that the conclusions arrived at in Proposition A hold good, no 

 matter what may ba the forms of the guide-tubes, provided that they be 

 similar in both flows ; and no matter what may be the distribution of 

 pressures throughout a terminal interface crossing the assemblage of guide- 

 tubes in No. 1, provided that the homologous pressures throughout a 

 homologous terminal interface in No. 2 be anyhow maintained severally n^ 

 times those in No. 1. Hence, if in Flow No. 1 the guide-tubes be formed so 

 that the water shall flow along exactly the same paths as if it were left 

 unguidcd, and were left free to shoot away, past the interface at E, to a 

 distance from the orifice great in proportion to the thickness of the issuing 

 stream, without meeting any obstniction — and if the guide-tubes in No. 2 be 

 similar to them — and if in No. 1 the system of pressures distributed through- 

 out the terminal interface at E be made exactly the same as if the water 

 were flowing freely for a great distance past that terminal intei-face — and if 

 in No. 2 the system of homologous distributed pressures throughout a homo- 

 logous terminal interface at E' be anyhow maintained severally n^ times those 

 in No. 1, — it foUows that the diff'erential pressure on the two sides of any 

 element of a septum in Flow No. 1 will be zero, as the guide-tubes have 

 there no duty to perform. Then, on the homologous septum element in 

 No. 2, the differential pressure, being n' times that in No. 1, will be zero 

 also. Hence in No. 2 the guide-tubes have no duty to perform, and the 

 water flows in them exactly as if it were left unguided, but had still through- 

 out its terminal interface the stated system of distributed pressures somehow 

 applied. 



Now, for completing the demonstration of Theorem II., nothing remains 

 needed except to show that this stated system of distributed pressures 

 requisite to be applied throughout the terminal interface at E' will very 

 exactly be applied on that interface backwards by the water in front of it, 

 which constitutes, for the time being, the continuation of the stream past 

 that interface. 



For proof of this, conceive any cross interface F F (fig. 9) further for- 

 ward in No. 1 than E E is, and conceive a similarly situated cross interface 

 F' F' in No. 2. By exactly the same mode of reasoning as before (making 

 use of the like supposed introduction and subsequent removal of guide-tubes), 



