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PROCEEDINGS OF THE AMERICAN ACADEMY. 



within and on the circumference of C, and by theorem IV is harmonic 

 within C. Accordingly its average on the circumference of C, which 

 is obviously given hy (2), is equal by III to its value at P', that is to the 

 value of u at P. Thus our theorem is proved. 



Formula (2) is merely an unfamiliar form of Poisson's Integral. 4 

 This may readily be shown by means of the following lemma, whose 

 extremely simple proof we suppress : 5 



Lemma ; If from a point P within a circle C two arcs are drawn 

 orthogonal to C and meeting it at A and A', the angle between these arcs 

 is measured by the arc BB' of C intercepted between the straight lines 

 AP and A'P produced. 



Applying this to the case in which AA' is the element of arc ds, the 

 angle between the two arcs PA and PA' will be d6. Denoting the arc 

 BB by ds u and the radius of the circle by R, this lemma tells us that 

 C&! = RdO. Accordingly (2) reduces to 



(3) 



— -£ J u a ds u 



the integral being extended around the circle and u s denoting the same 

 thing as u r This is the form of Poisson's Integral given by Schwarz 

 on p. 360, Vol. 2, of his collected works. 



4 See a Note on Poisson's Integral, by the writer, Bull. Amer. Math. Soc., June, 

 1898, where a somewhat different treatment is given. 



5 See a paper by the writer on Fourier's Series in the Annals of Mathematics for 

 January, 1906. 



