NOVEMBEE 11, 1898.] 



SCIENCE. 



643 



out viscosity and its motion is irrotational, 

 steady and takes place in two dimensions. 

 If we take the plane of motion to be that of 

 (x, y, ) this class, of course, includes the case 

 of motion in three dimeasions where the 

 motion of every particle in a straight line 

 perpendicular to the plane of (x, y) is the 

 same. The theory depends on the velocity- 

 potential <p and the steam-function 4'. If 

 u, V be the velocities of a particle of the 

 fluid at the point {x, y) the following 

 equations hold : 



ay 



dx' 

 Putting 



'dx ' 

 92/ ' 



z == x + iy 



v= + 



aj,' 

 a^ 



Zx ' 



ay 



+ 



w == y>+ ill', 



these equations show that the determina- 

 tion of the motion reduces to the discovery 

 of a function 



■which will satisfy the given boundary con- 

 ditions. 



When w has been determined as a func- 

 tion of z, the curves, 



w = pure imaginary, 



give the stream-lines, that is, the lines fol- 

 lowed by the molecules of the fluid in their 

 m.otion. The velocities are determined by 

 partial difierentiation with respect to x, y. 

 It is unnecessary to say more about the 

 general problem. It is that of Dirichlet 

 in the Theory of Functions. Solutions cau, 

 in general, only be obtained by the inverse 

 process of taking known solutions of 



3V 4. 9V . ^^ av , av- . 



dy ox ay 



dx 



and inquiring what boundary conditions 

 they cau be made to satisfy. When the 

 motion is discontinuous the boundaries, in- 



stead of being fixed walls, are free surfaces 

 along which the velocity is constant. 



As we are searching for the stream-lines 



w = pure imaginary, 



that is, the lines parallel to the real axis in 

 the w-plane, all that is necessary is to find 

 the curves in the a-plane which correspond 

 to lines parallel to the real axis in the 

 t«-plane. The problem is, therefore, re- 

 duced to one of conform representation. 

 For walls consisting of straight lines a few 

 cases have been long known, being solved 

 by Schwarz's method. Some new ones, 

 with more complicated rectilinear boun- 

 daries, were given by Rethy in 1895. 



In 1890 Mitchell gave a new form to 

 Schwarz's method which enabled him to 

 solve some further problems in discontinu- 

 ous motion. These are mainly the cases of 

 jets issuing from apertures into fluid which 

 may be at rest or in motion. This was 

 shortly afterwards modified and extended 

 by Love. Still later B. Hopkinson added 

 the case where several sources or sinks 

 might be present. (These are points where 

 the fluid is supposed to enter or leave in 

 finite quantities. ) The transformation does 

 not in general contain infinities, but Hop- 

 kinson includes the case where -p- becomes 

 az 



infinite like 



+ ■■■ + 



A. 



(z - «i)"" 



Several hydrodynamical exanjples are given, 

 but the most interesting are the electrical 

 applications. 



These problems are of little practical 

 value. They frequently demand the exist- 

 ence of dead water behind the moving 

 solid. For example, if a rectangular board 

 be drawn through a fluid it is directly seen 

 that the fluid behind the obstacle is far 

 from being at rest, which the theory would 

 indicate. Again, if water is entering into 



