260 Methods for the Solution of Special Problems [CH. vin 



a small linear element QQ' in the W -plane. If OP, OP' represent the values 

 z, z -f dz respectively, then the element PP' will represent dz. Similarly the 



dW 



element QQ' will represent d W or y- dz. 



dz 



Hence we can get the element QQ' from the element PP' on multiplying 



it by -y , i.e. by - < (z), or by fi (x 4- iy). This multiplier depends solely 

 dz oz 



on the position of the point P in the 2-plane, and not on the length or 



dW 

 direction of the element dz. If we express -y or $' (x + iy) in the form 



= <f)'(x + iy) = p (cos x + i sin %), 



we find that the element d W can be obtained from the corresponding element 



dW 



dz by multiplying its length by p or 



dz 



, and turning it through an angle 



%, or arg ( r ) . It follows that any element of area in the ^-plane is repre- 



\ dz J 



sented in the "FF-plane by an element of area of which the shape is exactly 

 similar to that of the original element, the linear dimensions are p times as 

 great, and the orientation is obtained by turning the original element through 

 an angle ^. 



From the circumstance that the shapes of two corresponding elements in 

 the two planes are the same, the process of passing from one plane to the 

 other is known as conformal representation. 



313. Let us examine the value of the quantity p which, as we have 

 seen, measures the linear magnification produced in a small area on passing 



from the ^-plane to the TF-plane. 



dW 

 We have p (cos % + i sin %) = -y = (/>' (x 4- iy) 



~ dx dx 



'by dx ' 



7\V %V 

 so that p = 



The quantity p, or 



dW 



dz 



<9F^ 



= A / I 



dx 



, is called the "modulus of transformation." 



We now see that if V is the potential, this modulus measures the electric 



intensity R, or A/f-rrj -f-(-^r-) . Since R = 4t7rcr ) this circumstance pro- 



V \ox J \ oy I 



vides a simple means of finding cr, the surface-density of electricity at any 

 point of a conducting surface. 



