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, 0, we have 



&amp;lt;p = Ar n cos n0, \ 



^ = Ar n smn0 j * 



The following cases may be noticed. 



1. If ?i = l, the stream-lines are a system of straight lines 

 parallel to x t 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 (/&amp;gt; = const, are a system of rectangular 

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

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

 ordinate axes as asymptotes. The lines 6 = 0, \ IT 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 &amp;lt;p = A/r.cos 0, the 

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

 region to which our formulas 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 &amp;lt; = const, coincide 

 with x or y ; those of the system -fy = 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 sin nO = 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. 



