496 MI!. II. A. WEBB ON THE CONVERGENCE OF 



*When ||< 1, 



n, l-p-n, l-m, z) = n(-).(Y 



and the equality still holds when \z\ > 1, if F(p + n, 1 pn, \m, z) be replaced by 

 the function derived from it by analytic continuation. 



Tln'orem IX. Let <f>(z) be any function of z which is regular at all points in the 

 interior of an ellipse C, whose foci are at the points z = I and z = 1. 



The ellipse is so large that its circumference passes through one (or more) of the 

 singularities of <f>(z). The curve is thus completely defined when (f>(z) is given. 



Let m and p l>e any constant quantities whatever, real or complex. 



Then <ft(z) can be expanded in the infinite series of LEGENDRE'S associated functions 



% +1 (e)+...+.%,P"V + (-)+--] (36). 



where 



. ^ D (p-m+n) f (1+ ' ~ 1+i 1 ~- ''"'/I + A 1 " n 



= *(2 + 2n+ 1) - r -I P v + n (t)<b(i) . at. 



n(p + m + n) j \lt/ 



The series is convergent if z is inside C and divergent if z is outside C. If z is on 

 C, the series is, in general, oscillatory and the expansion fails. 



Theorem X. If the point p. is in the interior of the ellipse which passes through p, 

 and has the points fi = 1, /A = 1, for foci, and if m is any constant quantity what- 

 ever, real or complex, 



can be expanded in the infinite series of LEGENDRE'S associated functions, 



-( /B )+...} (37). 



If p. is outside the ellipse, the series is divergent. If fi is on the ellipse, the series 

 is, in general, oscillatory and the expansion fails. 



Theorem XL Let <(z) be any function which is regular at all points in the 

 interior of an ellipse C, whose foci are at the points 2 = 1 and z 1. 



The ellipse is so large that its circumference passes through one (or more) of the 

 singularities of <(z). The curve is thus completely defined when (f>(z) is given. 



Let m be any constant quantity, such that the real part of m lies between 

 1 and -1. 



* HORSON, "On a Type of Spherical Harmonics of unrestricted Degree, Order and Argument," 'Phil. 

 Trans.,' 1896, Series A, vol. 187, p. 451 (5). The notation used by Honsox, including the definitions of 

 the functions P m (/*), Q n m (p.), will lie adopted. 



