578 PROCEEDINGS OF THE AMERICAN ACADEMY. 



' 9u 



I 



, ds=0 



dn 



the integral being extended around the circle, n denoting the exterior normal 

 and s the arc. 



It is clear that this includes all functions which are harmonic according 

 to the ordinary definition. It is not so clear that it does not also include 

 other functions ; for since we have not demanded even the existence of 

 the second derivatives of u, we cannot pass from (1) to Laplace's equa- 

 tion by the familiar application of Green's theorem. 



Theorem I. On the circumferences of any two concentric circles 

 lying wholly in T the average values of u are the same. 3 



Call the radii of the circles a and b (« < b) and use polar coordinates 

 (r, cf>) with pole at the centre of the circles. Then by our definition 



Since 9u/3r is a continuous" function of (r, <£) we may reverse the order 

 of integration. Denoting by u a and u b respectively the values of u on 

 the smaller and larger circle, we thus get : 



X 



277 



[«& — « a ] d$ = 0. 







from which our theorem follows at once. 



By allowing a to approach zero as its limit we get 



Theorem II. The average value of u on the circumference of any 

 circle lying wholly in T is the value of u at the centre of this circle. 



This theorem we next state in the following generalized form : 



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

 of a circle C, and is harmonic within it, then the average value of u on 

 the circumference of C is its value at the centre of C. 



For the average value of u on the circumference of a circle a little 

 smaller than C and concentric with it will, on account of the continuity 



8 The proofs here given of this theorem and the next were used by the writer 

 in a paper on Gauss's Third Proof of the Fundamental Theorem of Algebra. Bull. 

 Amer. Math. Soc, May, 1895. 



