1891.] Fluid Motions in two dimensions. 177 



boundary. In this way there will arise sufficient relations to 

 determine all the arbitrary constants. If we integrate the equation 

 connecting z and u for values of u that correspond to a free stream- 

 line we shall find for z a complex expression, so that the co- 

 ordinates x and y of any point on the free stream-line will be 

 given functions of a real parameter u. 



Mr Michell's theory rests on the properties of the function 

 which is the real part of XI. He shows how to determine this 

 function in terms of u by considering an analogous electrical 

 problem. When it is known he deduces from it the differential 

 relation between z and u that would be found by the method of 

 this paper. 



After explaining the method I give solutions of the following 

 problems : 



(i) Mr Michell's problem of the escape of a jet from a tank. 



(ii) The flow of liquid against a disc with an elevated rim. 



(iii) The impact of a jet against a finite lamina. 



(iv) The resistance offered by a plane obstacle placed in a 

 canal of finite width. 



(v) The flow of liquid past a pier projecting obliquely. 



1. The theory of discontinuous fluid motions in two dimensions, 

 as at present developed, rests essentially on two particular pro- 

 positions to which we proceed. 



Prop. I. The motion being supposed to take place in the 

 plane of the complex variable z, and to be given by means of a 

 velocity-potential <£ or a stream-function ty, so that 



w = <J> + iylr=f (x + iy) =f(z), 



the quantity dz/dw, which we call £, is a complex variable whose 

 modulus is the reciprocal of the velocity, and whose argument is 

 the direction of the velocity of the fluid at the point z ; and the 

 quantity log £, which we call fl, is a complex variable whose 

 real part is the logarithm of the reciprocal of the velocity and 

 imaginary part is V( — 1) multiplied by the angle the direction of 

 the velocity makes with the real axis in the z plane. 



To prove this observe that if u, v be the velocities at any point 

 parallel to x, y, then 



dw dw dd> d-fr 



dz ox ox ox 

 ,i „ dz u + iv u + iv 



SOthat ^ = dw = u^7 == ~^- S ^ 



and fl = logr=log^ 2 -=logi + ^ (A), 



where $ is such that cos 6 = u/q and sm0 = v/q. 



