43, 44] FUNCTIONS OF A COMPLEX VARIABLE. 85 



and therefore 



cZcTj : dff z : da- 3 ds 1 : ds 2 : ds 3) 



and the infinitesimal triangles are similar. Consequently corre- 

 sponding curves intersect each other in the same angle in both 

 corresponding planes. Such a relation as this is called a Conformal 

 Relation*, and it is of fundamental importance in the theory of 

 functions and in mathematical physics. The two planes are said 

 to be conformal representations of each other. The relation is 

 sometimes specified by saying that the conformal representations 

 are similar in their infinitely small parts. 



It is easy to show that if the functions u and v give a conformal 

 representation of the plane, they must satisfy the equations (A ). 



44. Laplace's Equation. Conjugate Functions. If we 



differentiate the equations (A), the first by x and the second by y 



d*v d*v 

 and add, since ... ... = . . we have 



Gxoy oyox 



so that the function u satisfies Laplace's equation in two variables, 

 or is harmonic. 



Differentiating the other way and adding we show that v also 

 satisfies the same equation 



Thus every conformal development or every analytic function 

 of a complex variable gives us two harmonic functions. The 

 question arises whether the converse is true. It obviously will 

 not do to take any two harmonic functions for u and v, for they 

 must be related so as to satisfy the equations (A). But if one 

 function is given, we may find the conjugate, for we must have 



, dv , dv j 

 dv = 5- dx + 5- dy, 

 das dy ' 



which by the first equation (A ) is 



, du j du j 

 cfo; _ dx + ^- dy. 

 dy das 



* Professor Cayley has called it orthomorphosis. 



