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



of the velocities on the two sides should differ by a constant 

 quantity. If further, as occurs in most cases of practical in 

 terest, the fluid on one side of the surface be sensibly at rest, 

 the conditions reduce to these, that the velocity must be wholly 

 tangential to the surface, and of constant magnitude. 



Kirchhoff s Method. 



95. Kirchhoff has applied the method already partially ex 

 plained to obtain forms of discontinuous motion (in two dimen 

 sions) satisfying the conditions just stated. The fluid is supposed 

 bounded by two stream-lines i/r = a, ^r = /5, consisting partly of 

 fixed boundaries and partly of lines of constant (say unit) velocity. 

 The lines for which q 1 are represented in the plane of f by 

 arcs of a circle of unit radius having the origin as centre ; whilst 

 the fixed boundaries, if straight (as we shall suppose them to be), 

 become radii of this circle. The points where the radii meet 

 the circle correspond to the points where the limiting stream-lines 

 change their character. We have then to frame an assumption 

 of the form 



such that the portion of the plane of external to the above 

 circle and included between the two radii shall correspond to the 

 portion of the plane of w included between the two parallel 

 straight lines ^r = a, ty = fi. It is further necessary (see Art. 79), 

 that the function f(w) shall have no branch-points within the 

 portion of the plane of w considered, although such points may 

 occur on the boundary of this portion. 



It is found that this problem may be reduced to a particular 

 case of the following : To connect two complex variables z y z 

 by a functional relation such that any given lune in the plane 

 of z shall correspond to any given lune in the plane of z\ 

 and any three points on the perimeter of the one lune to any 

 three points on that of the other. By a lune is here meant 

 the closed figure formed by two circular arcs which meet but 

 do not cross. By the angle of a lune we shall understand the 

 angle contained by the arcs at either intersection. 



