310 ELECTROSTATICS. [PT. II. CH. VII. 



We now require the condition that an equation </> (#, y)=G 

 represents an equipotential family of curves. In this case we shall 

 have for the potential function F, F=/(^>) and as in 108, (2) 



(6) 



so that 



/'(</>)' 





The right-hand member depends on </> alone, consequently the 

 left hand must also. Consequently in order that < (a?, y)= C shall 

 represent an equipotential family the ratio of the second to the 

 square of the first differential parameter of must be a function 

 of <f> alone. Let now c/> (x, y)=C represent an equipotential family, 

 and let <I> (u,v) = C' be the transformed family. Since by (5) 



V 2 ~ V ' 



and since A<//^ 2 depends only on cf>, A'<J>/^ /2 will depend only on 

 <E>, for <j> and <E> are constant together. 



Accordingly a conformal transformation leaves every equipo- 

 tential family equipotential. It is upon this property that the 

 application to electrostatical and other physical problems depends. 



If we integrate the second parameter of F over a portion of the 

 JTF-plane where it does not vanish, using the element of area in 

 curvilinear coordinates 



dudv 



/ Q\ I I I v r . ** ' X 7 i I I T / I/ r . V f \ CtUCtV 



() 



and now considering the second integral to refer to the trans- 

 formed plane, and e and e' to be charges of corresponding regions, 



(9) e=jj p dxdy=-^H&Vdxdy = -^ ff XV dudv 



= II p dudv e', 



or corresponding regions in the two planes have equal charges (the 

 densities being different). 



