BOCHER. — HARMONIC FUNCTIONS IN TWO DIMENSIONS. 579 



of u, approach the average value of u on the circumference of C as the 

 smaller circle approaches as its limit. From this fact and Theorem 

 II our theorem follows. 



Theorem IV. If u is harmonic within T, and the region T is 

 carried over by inversion into a region T' ; and if we define a new func- 

 tion u' in T' so that it has the same value at every point of T' that u has 

 at the corresponding point of T then u' is harmonic in T '. 



In the first place, inversion being an analytic transformation, u' is 

 evidently continuous throughout T 1 and has continuous first partial 

 derivatives there. Now consider any circle C lying wholly in T', and 

 denote by s' the arc and by n' the exterior normal of this circle. If G 

 is the circle in T of which O is the inverse and we consider correspond- 

 ing points on these two circles, we have, since inversion is a conformal 

 transformation, after neglecting infinitesimals of higher order : 



ds' ds 



dn' dn ' 

 Accordingly 



M> dS ' = fPn ds = °> 



the first integral being taken around C, the second around C. This 

 completes the proof that u' is harmonic. 



Theorem V. If u is continuous within and on the circumference 

 of a circle C, and is harmonic within C, then the value of u at any point 

 P within C is given by the integral : 



1 p8-2n 

 (2) 5-/«.^, 



where denotes the angle between a fixed and a variable circular arc 

 which both start from P and which both cut C at right angles, and \\q de- 

 notes the value ofuat the point where this variable arc cuts C. 



In order to prove this, let us construct the point Q inverse to P with 

 regard to C. If we invert the whole figure with regard to a circle having 

 Q as centre, P goes over into a point P\ C into a circle C having P' 

 as centre, and the fixed and variable arcs referred to in the theorem go 

 over into fixed and variable radii of C making the variable angle 6 with 

 one another. Finally, we define a function u' within and on the circum- 

 ference of O so that it has the same value at every point that u has at 

 the corresponding point of C. This function u' will then be continuous 



