18 REPORT— 1870. 



In the same way there is a system of watersheds forming the hoimdary of a region 

 called a Dale, within which all the lines of slope run to the same bottom. 



The whole sui-face of the earth may be divided into Hills, the number of these 

 being the same as that of their Tops. 



By an independent division, the whole siu-face may be divided into Dales, each 

 Dale having a different Bottom. 



Besides this, we may, by superposing these divisions, consider the earth aa 

 divided into Slopes, each slope being bounded by two watersheds and two water- 

 com'ses, and being named from the top and the bottom between which all its lines 

 of slope run. 



The number of Slopes is shown to be equal to the total number of Tops, Bottoms, 

 Passes, and Bars minus two. 



An Investigation of the MatJiematical TJieory of Combined Streams. 

 By W. J. Macqtjorh Eaotone, C.E., LL.D., F.R.SS. L. Sf E* 



_ The object of the investigation, of which this is an abstract, is to extend to com- 

 binationsof any number of streams of fluid, whether liquid, vaporous, or gaseous, 

 the principles which have been applied to combinations of two streams by previous 

 authors, and especially by Professor Zeuner, in his treatise entitled " Das Locomo- 

 tivenblasrohr " (Ziirich, 1863). Several component streams of fluid, each coming 

 through its own supply-tube and nozzle, are led in directions pai-allel to each other 

 into one end of a cylindrical space called the jimction-chamber, in which they 

 mingle so as to form a resultant stream ; and that resultant stream escapes from 

 the other end of the junction -chamber through an orifice called the throat. The 

 dynamical principle upon which the motion depends is that of the equality of 

 impulse and momentum. The aggregate momentum per second of the component 

 streams is found by midtiplying the mass of fluid which comes from each nozzle in 

 a second by its velocity, and" adding together the products. The momentum of the 

 resultant stream is the product of the mass of fluul discharged from the throat in a 

 second, into the velocity at the throat. The diflerence of these two momenta is 

 equal to the impulse per second exerted in the junction -chamber, which impulse is 

 found by multiplying the area of the throat by the difference between the inten- 

 sities of the pressure at the nozzle-end and at the throat-eud of the chamber respec- 

 tively. If there is a gain of momentum, the pressure at the thi-oat is less than at 

 the nozzles ; if there is a loss of momentum, tne pressure at the throat is greater 

 than at the nozzles. 



There is always a loss of energy, which is expended in producing eddies, imlesa 

 the velocities of the component and resultant streams are equal to each other. 

 The amount of that loss can be calculated in any given case by the help of the 

 principle already stated ; and that principle being expressed in the fonn of an 

 equation, and taken together with another equation expressing the equality of the 

 mass discharged at the throat to the sum of the masses which come through the 

 nozzles, affords the means of solving various problems as to combined streams. 



1 



On the Tliermodynamic Acceleration and Retardation of Streams. 

 By "W. J. MAcaxjOEN Eankine, O.E., LL.D., F.R.SS. L. 6f E.f 



The object of this paper is to state in a more general and comprehensive form 

 than has hitherto been done to my knowledge, a thermodynamic and hydi'odyna- 

 mic principle of which many particular cases are well known and understood. 

 That principle may be stated as follows : — 



In a steady stream of am/Jlia'd, the abstraction of heat at and near places of mint- 

 mum 2n-essure, and the addition of heat at ami near phtces of maxintiim pressure, tend 

 to produce acceleration ; the addition of heat at and near jilaces of minimum pressure, 

 and the abstraction of heat at and near places of maximum pressure, tend to produce 



* Printed in full in the Proceedings of the Eoyal Society, 1870, No. 123. 

 t Printed in full in the Philosophical Magazine for October 1870. 



