42, 43] FUNCTIONS OF A COMPLEX VARIABLE. 83 



43. Orthogonal Coordinates. Conformal Representation. 



We may apply the considerations of 15 and 20 to the case of 

 orthogonal coordinates in a plane. If a set of point-functions are 

 independent of one rectangular coordinate, the geometry of all 

 planes perpendicular to the axis of that coordinate is the same, 

 and we have the uniplanar, or two-dimensional case involving only 

 two variables which we will take as x, y. If we take u and v as 

 any two point-functions, whose parameters are h u , h v 



"<u = U- +15- > V "* 1 2T ) + VST 1 > 

 \dxj \oy/ \dxj \dyj 



their level lines u = constant and v = constant may be taken for 

 coordinate lines. 



Their normals have the direction cosines 



1 du 1 du 



l Sv 



and the condition that u and v shall form an orthogonal system is 



du dv du dv _ - 

 dx fix dy dy 



The lengths of infinitesimal arcs of curves, forming the sides 

 of a rectangle whose opposite vertices have coordinates u, v, u + du, 

 v + dv. are as in 20 



du dv 

 h u ' h v } 



and the length of the diagonal ds, or element of length of a curve 

 whose ends have the above coordinates, is given by 



du 2 dv 2 



ds = n + O ' 

 A M 2 hf 



If now we take for curvilinear coordinates in the #, y plane two 

 functions u and v such that u + iv is an analytic function of as + iy, 

 in virtue of the equations (A) of 42 we have 



du dv du dv _ - 

 dx dx dy dy 



and u and v form an orthogonal system. Now in any orthogonal 

 system if we construct a set of level curves for equal small incre- 

 ments of u and v, they will divide the plane up^into small 



(( UNIVE! 



