78 



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



angular co-ordinates of the point P in the plane of w corresponding 

 to the point P (x, y] of the plane of z. If P describe a circle of 



Fig. 5. 



w 



radius unity about the origin, starting from the point (1, 0) ; then, 

 since log 1 = 0, P will move along the axis of ty, and will describe 

 a length 2?r of that axis for every revolution of P . To points P 

 outside the circle r = 1 correspond points P to the righ t of the 

 axis of i/r ; and vice versa. To every straight line through the 

 origin, in the plane of z, corresponds in the plane of w a straight 

 line parallel to the axis of &amp;lt;. The periodicity of z t and the cyclosis 

 of w, are manifested by the division of the plane of w by the 

 straight lines ^ = 2?i?r into an infinite number of compartments, 

 each of which corresponds to the whole of the plane of z. 



In the same way the properties of the function arc tan z may be 

 deduced from the definition 



arc tan z = 



dz 



z i 



and, though with much greater difficulty, those of the function 

 arc sin z from the definition 



arc sin z 



C z dz 



= I 



Jo s /l- 



