304-308] Conjugate Functions 25 1 ? 



then F(x -f iy) must be equal to u iv, so that we must have F = 2w. If we 

 introduce a second function U equal to 2v, we have 



= <t>(x + iy) ........................ (230), 



where <f> (x + ^y) is a completely general function of the single variable x + iy. 



Thus the most general form of the potential which is wholly real, can be 

 derived from the most general arbitrary function of the single variable x + iy, 

 on taking the potential to be the imaginary part of this function. 



307. If < (# + iy) is a function of x + iy, then i(f> (x + iy) will also be a 

 function, and the imaginary part of this function will also give a possible 

 potential. We have, however, from equation (230), 



shewing that U is a possible potential. 



Thus when we have a relation of the type expressed by equation (230), 

 either U or F will be a possible potential. 



308. Taking F to be the potential, we have by differentiation of equa- 



tion (230), 



8Z7 . dV .,. 

 --M-^^ + ty), 



dU .9F 



_+t =t*(*+ty), 



.ftu .aF\ dU .dv 



and hence *r^~ + *3~ =: ^~+ l ^~- 



\dx ox J oy dy 



Equating real and imaginary parts in the above equation, we obtain 



d_U_dV 

 dx " dy ' 



sothat 



This however is the condition that the families of curves 7= cons., 

 F=cons., should cut orthogonally at every point. Thus the curves 

 t/" = cons., are the orthogonal trajectories of the equipotentials i.e. are the 

 lines of force. 



j. 17 



