197] 



CONJUGATE FUNCTIONS. 



525 



whose derivative is gp + iif>. From these by new integrations we 

 may obtain any number. The method of the complex variable accord- 

 ingly gives us the solution of an unlimited number of uniplanar 

 problems. 



The equations 122) are geometrically the condition that the lines 

 (p = const., iff = const, intersect each other everywhere at right angles. 

 If ijj is the stream function the lines ijj = const, are the lines of flow, 

 which we know intersect the equipotential surfaces at right angles. 

 As examples consider the cases worked above, 



w = 2 , (p = x 2 2/ 2 , $ = %xy. 



The equipotential lines are sets of equilateral hyperbolas, intersected 

 at right angles by the system of equilateral hyperbolas forming the 

 stream lines (Fig. 164). The stream line iff = consists of the X and 

 Y axes, which may accordingly be a boundary, so that one quarter 



Fig. 164. 



of the figure represents the flow in a square corner of a stream of 

 infinite extent. 



The function w = gives 



x .1. _ y 



