BOCHER. — HARMONIC FUNCTIONS IN TWO DIMENSIONS. 581 



This form may be still further changed by means of the proportion 

 dsi/pi = ds/p, where p\ = PB, p = PA. It may thus be written : 



VrJ u ' 



P -^ds. 



Now, p x p = (R — r)(B + r), where r denotes the distance from P to 

 the centre of G. Thus, finally, (2) takes the form: 



1 f J^ _ r -2 

 W 2^Rj U '—f- ds > 



which is Poisson's original form. 



Since (R' 2 — r 2 ) /p' 2 is an analytic function of the coordinates {x, y) of 

 P, the same will be true of the whole integral (4), and therefore u(x,y) is 

 analytic throughout the interior of the circle G. Moreover, (R 2 — r' 2 ) /p' 2 

 may be seen, by direct differentiation or otherwise, to satisfy Laplace's 

 equation. Hence we infer that the same is true of the function u(x, y) 

 represented by (4). 



We have been assuming that u is harmonic merely within G. If, 

 however, it is harmonic throughout any region T, we can now infer at 

 once that it is analytic at any point P of this region, and satisfies 

 Laplace's ecmation there by merely surrounding P by a small circle 

 and applying the results just established to this circle. Thus we have 

 proved 



Theorem VI. Ifn (x, y) is harmonic within any region T, it is ana- 

 lytic and satisfies Laplace's equation at every point within T. 



We have thus established the identity of the class of functions defined 

 above as harmonic with the class ordinarily so defined. 



In conclusion I add the following remarks on the result thus obtained. 



1) In the definition of a harmonic function it is obviously not necessary 

 to demand that (1) hold for all circles contained in T. All that is essen- 

 tial in the reasoning we have given will apply at once if we start from the 

 following definition : 



The function u (x, y) is said to be harmonic at a point (x x , y^, if it is 

 possible to surround this point by a region so small that u is continuous 

 and has continuous first partial derivatives there, and for every circle 

 contained in this region satisfies equation (1). The function u is said 

 to be harmonic throughout a region T if it is harmonic at every point 

 of this region. 



