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



the real axis of w, drawn on the negative and positive sides, respectively*. 

 The reader should examine these statements, as we shall have repeated occasion 

 to use this transformation. 



63. We can now proceed to some applications of the foregoing 

 theory. 



First let us assume w = Az n , 



A being real. Introducing polar co-ordinates r t 0, we have 



The following cases may be noticed. 



1. If n = 1, the stream-lines are a system of straight lines 

 parallel to #, and the equipotential curves are a similar system 

 parallel to y. In this case any corresponding figures in the planes 

 of w and z are similar, whether they be finite or infinitesimal. 



2. If n = 2, the curves <f> = const, are a system of rectangular 

 hyperbolas having the axes of co-ordinates as their principal axes, 

 and the curves ty = const, are a similar system, having the co- 

 ordinate axes as asymptotes. The lines 6 = 0, = \ TT are parts of 

 the same stream-line ty = 0, so that we may take the positive parts 

 of the axes of %, y as fixed boundaries, and thus obtain the case of 

 a fluid in motion in the angle between two perpendicular walls. 



3. If n = 1, we get two systems of circles touching the 

 axes of co-ordinates at the origin. Since now < = A/r . cos 0, the 

 velocity at the origin is infinite ; we must therefore suppose the 

 region to which our formulae apply to be limited internally by a 

 closed curve. 



4. If n = 2, each system of curves is composed of a double 

 system of lemniscates. The axes of the system < = const, coincide 

 with x or y ; those of the system ty const, bisect the angles be- 

 tween these axes. 



5. By properly choosing the value of n we get a case of 

 irrotational motion in which the boundary is composed of two 

 rigid walls inclined at any angle a. The equation of the stream- 

 lines being 



r n sinn0= const ......................... (2), 



* It should be remarked that no attempt has been made to observe the same 

 scale in corresponding figures, in this or in other examples, to be given later. 



