102 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



A familiar instance is the smoke-laden stream of gas issuing from 

 a chimney. 



74. Leaving aside the question of the manner in which the 

 motion is established, von Helmholtz* and Kirchhofff have 

 endeavoured to construct types of steady motion of a frictionless 

 liquid, in two dimensions, which shall resemble more closely what 

 is observed in such cases as we have referred to. In the problems 

 to be considered, there is either a free surface or (what comes to 

 the same thing) a surface of discontinuity along which the moving 

 liquid is in contact with other fluid at rest. In either case, the 

 physical condition to be satisfied at this surface is that the 

 pressure must be the same at all points of it; this implies, in 

 virtue of the steady motion, and in the absence of extraneous 

 forces, that the velocity must also be uniform along this surface. 



The most general method we possess of treating problems of 

 this class is based on the properties of the function f introduced 

 in Art. 65. In the cases we shall discuss, the moving fluid is 

 supposed bounded by stream-lines which consist partly of straight 

 walls, and partly of lines along which the resultant velocity (q) 

 is constant. For convenience, we may in the first instance suppose 

 the units of length and time to be so adjusted that this constant 

 velocity is equal to unity. Then in the plane of the function 

 f the lines for which q = 1 are represented by arcs of a circle 

 of unit radius, having the origin as centre, and the straight 

 walls (since the direction of the flow along each is constant) by 

 radial lines drawn outwards from the circumference. The points 

 where these lines meet the circle correspond to the points where 

 the bounding stream-lines change their character. 



Consider, next, the function log f. In the plane of this 

 function the circular arcs for which q = 1 become transformed into 

 portions of the imaginary axis, and the radial lines into lines 

 parallel to the real axis. It remains then to frame an assumption 

 of the form 



log ?=/(>) 

 such that the now rectilinear boundaries shall correspond, in the 



* 1. c. ante p. 24. 



t "Zur Theorie freier Flussigkeitsstrablen, " Crelle, t. Ixx. (1869), Ges. Abh., 

 p. 416; see also Mechanik, cc. xxi., xxii. Considerable additions to the subject 

 bave been recently made by Michell, " On the Theory of Free Stream Lines," 

 Phil. Trans., A., 1890. 



