tss MR. II. A. WEBB ON THE CONVERGENCE OF 





 where the C'B are arbitrary, save that the series 2 c m z" has unit radius of convergence, 



ml 



converges if 



To interpret this condition geometrically, let 



x = rc io , t = pe~, 

 and the condition reduces to 



......... (15). 



7. *If we are given any finite simply-connected plane area whose boundary either 

 is an analytic curve or is made up of portions of a finite number of analytic curves, 

 the interior can be conformally represented by the interior of a circle, any given point 

 corresponding to the centre of the circle, and the equation of the boundary can be 

 expressed in Cartesian co-ordinates (f, 17) in the form |0(z)| = constant, where 

 z = +ir), and 6(z) is a uniform analytic function of z. 



The following is a converse of Theorem II. 



TJieorem III. We can construct a series, F(x, t), which converges if x lie within 

 the area bounded by 1 6(z) \ = \0(t)\, but not if re lie outside the area. 



Points lying on the boundary of the area are excluded. 



(16), 



where c,, c 2 ...c n . are arbitrary, save for the restriction that S C B Z" has unit radius of 



n=l 



convergence, and f n (z), g n (z) are solutions of the linear differential equation of the 

 second order 



where r, a are real positive constants, independent of n, f^(z),f^(z}... are functions of 2 

 regular within the curve 1 6(z) \ = \ 0(t) \ , and the series in the bracket is arranged in 

 descending powers of n and is convergent when n is large. 



8. Tfieorem IV. Let (f> (z) be a solution of the linear differential equation of the 

 w th order 



l y = 0. . . ..... '. . , (18), 



the coefficient of d r y/dz r being a polynomial in z of order r. 



* FORSYTH, 'Theory of Functions,' 2nd edition, chapter XX. 



