RECENT TROGBESS IN HYDRODYNAMICS. 69 



external forces through its mass, or if there be no fluid, the pressure 

 along the surface must bo constant, and, therefore, also the velo- 

 city. Helmholz illustrates this by considering the function x + yi = 

 A {^ + v//i + ex2) (</) + 4ii)} ■ This gives the lines of flow of electricity in 

 an infinite conducting sheet, in which two parallel non-conducting lines are 

 drawn stretching from the points (— A, + Att) to — go . It is, therefore, 

 the solution, so far as the equation of continuity is concerned, for fluid 

 flowing from between two infinite parallel walls into an infinite fluid ; but 

 the conditions at the mouth break up the fluid. He then, by adding 

 (T -H Ti, a function of ^ -f- \lii, to the former expression, determines ff, r, 

 so that when xp ■= ± w, the velocity may be constant along the free 

 surface. The value obtained is 



a + ri = Ai [\/ { — 2 exp f + 4^1 — exp 2 (f + \pi)} 

 + 2 sin ~^{i\/^ exp (<j> + '/'i)}]- 



It is then easy to show that the expression for x + yi gives the motion 

 from an infinite fluid into a canal with parallel walls, extending to infinity 

 in the negative direction. At a great distance from the entrance, in the 

 canal, the fluid tends to flow in a stream whose breadth is half that of 

 the canal. 



In Helmholtz' example it seems a happy chance by which a suitable 

 function is found. KirchhoS"' remedied this want of method, to some 

 extent, in a paper which contains several further examples of discontinuous 

 motion. Denoting by z the complex x + yi, and hy w, (p + xpi, the 

 conditions are — for the rigid boundaries, \p = constant ; and, for the free 

 boundary, \p = same constant, and the velocity constant. To this end he 

 puts 



and chooses/ (iv), so that for a certain value of \p, and a certain interval 

 of (j), f (w) is real, and lies between the limits + 1 and — 1. For these 

 values, then, 



or the velocity is constant, and equal to unity. The equation in dz/dw 

 will always give to as a many-valued function of z ; now the region of z 

 is the space occupied by the fluid, and since in this space one branch of 

 the function must not pass over into another, the region must be so 

 chosen that within it / (w) is single-valued, or, if it is not so, it must be 

 made so by cuts from the branch points to infinity, along \p = const., and 

 again v' {(/^)^ — 1} made single- valued by cuts from its branch points, 

 along the curves i// = const, through them. The boundary of the region of 

 IV consists, then, of lines \p = const., and (^ = — ooto^ = + oo, and within 

 it/ (w) must nowhere be infinite. In this case the lines i// will be stream 

 lines, parts of which form rigid walls, and the other parts free surfaces. 

 In his treatise on mechanics,^ he has introduced the subject in a slightly 

 different and improved manner. The dz/dw of the foregoing represents 

 the inverse velocity at the point z ; hence, along a straight rigid boundary, 

 c = dz/dw is constant in direction, whilst along a free surface it is 

 constant in magnitude ; in other words, as z moves along a bounding 



' ' Zur Theorie freier Fliissigkeitsstrahlen,' Crelle, Ixx. p. 289. 

 " Vorlesxmgen ilber Tnathcmafnche Physih. Leipzig. 



