158] CONFORMAL REPRESENTATION. 309 



Adding the second derivatives, and making use of the funda- 

 mental equations of conjugate functions, 42 (A), 44, 



we obtain 



If we call 



we see that in a conformal transformation the second differential 

 parameter A'F of any function V in the transformed plane is equal 

 to the second differential parameter AF for the corresponding 

 point in the original plane divided by the square of the ratio of 



dw 



_ . , 



" dx 



linear magnification h = 



dz 



at the point P' ( 43). Consequently 



a harmonic function of x, y is transformed into a harmonic function 

 of u, v. 



In like manner squaring the first derivatives and adding, we 

 obtain 



(Ming 



\daj \dvj 



we see that the square of the first differential parameter h v pos- 

 sesses the same property with regard to the transformation. 



Dividing equation (3) by (4) obtain 



CO ~^ = ^ = ~~ T^" 1 ^^?^" = -^ r > 



\J/ t 2 /^T7\2 /32 T7\ 2 /3 T7\ 2 /3 l/\ 2 7> 2 ' 



fly [0 V \ (Or\ (u v \ 10 V \ fly 



or the ratio of the second differential parameter of any function to 

 the square of the first is unchanged by a conformal transformation. 

 We may call such a quantity an invariant of the transformation. 



