By differentiating Equitions [22c,d] once more, it is found that 



^2, (92,/, ^2^ ^2^ 



Thus every regular function of z furnishes at once two real solutions of the Laplace equation 

 [7a] in two dimensions; they are obtained, respectively, from the real part and the imaginary 

 part of the function. This principle furnishes a powerful means of discovering such solutions. 

 Furthermore, as has been seen, the two families of curves, <^ = constant and ih = constant, 

 intersect everywhere orthogonally, as illustrated in Figure 29, except perhaps where dw/dz 

 vanishes or at a singular point. 



Obviously either <;6 or tA can be employed as the velocity potential for a type of 

 irrotational flow. 



If (f) is the potential, the x and y components of the velocity are 



w = , v L25a,b] 



dx dy 



Thus, using Equation [22c,d], 



u = , V = ^ [25c, d] 



dy dx 



also. The agreement of these equations with Equations [13a, b] shows that dj represents the 

 stream function as previously defined. 



The functions (p and t// have thus all of the properties of the conjugate functions de- 

 scribed in Section 13. The relationship is reciprocal; for, any solutions <f) and il/ of the two- 

 dimensional Laplace equation that satisfy Equations [22c,d] can be used to construct a reg- 

 ular transformation, w = (f> + i tp. Thus conjugate functions can be defined, as an alternative, 

 in terms of their relation to certain regular transformations of a complex variable. 



Each transformation yields two conjugate types of flow, In the second type, the velocity 

 potential 0'and stream function i/i'are related to <^ and ijj by the equations 0'= i/r, t//'= -<^, 

 and the components of the velocity are 



,_ ^'__^'__^ / ^'_^'_ ^ [OK n 



dx dy dx dy dx dy 



Use has been made here again of the Cauchy-Riemann relations. Equations [22c, d]. This 

 second type of flow can also be regarded as arising from the modified or conjugate transforma- 

 tion 



w' = <^' + i ip' = - iw =- if{3) = i/f - t [25g] 



Thus the conjugate flow is substituted for the original if in all formulas iw is substituted for 

 Wf since iw' = w. 



Comparison of Equations [25e,f] and [25c,d] shows, as stated in Section 13, that the 

 vector velocity in the second type of flow can be produced by rotating the velocity in the first 



47 



