The values of and of i/f that are thus associated with each point on the s-plane can 

 be employed as curvilinear orthogonal coordinates on that plane. Thus each regular transfor- 

 mation furnishes in and ifj a special set of orthogonal coordinates. 



\f w = f(z) is a many-valued function but such that any one branch of it, taken by itself, 

 has a unique derivative, then each branch maps an area of the 2-plane onto the w-plane, inde- 

 pendently of all other branches. 



Finally, a device pointed out by Maxwell may be mentioned that is sometimes useful in 

 drawing the curves. Suppose the curves, if/ = constant, are to be drawn, and that ip is the sum 

 of two terms: 



c^(a;,y) = f{x,y) + g{.x,y) 



First draw the two sets of curves, f{x,y) = constant, g{x,y) = constant, using the same equal 

 spacing for the constant values of / and g. These curves divide the plane into approximate 

 parallelograms. Then it is easily seen that curves, il/(x,y) = constant, for equally spaced 

 values of ip, pass through opposite corners of these parallelograms as illustrated in Figure 24, 



Figure 24 - Maxwell's construction for curves defined by the sum of two functions. 



24. EXAMPLES OF CONFORMAL TRANSFORMATIONS 



(1) Consider first the linear transformation 



w = Az + B 



where A and B are fixed numbers, perhaps complex. Let A = a + ib wher§ a and b are real. 

 Then 



42 



