494 MR. H. A. WEBB ON THE CONVERGENCE OF 



that is, according as x lies within or without an ellipse passing through t, which has 

 the points and 1 for foci. 



We have the following result. 



Theorem VI. If the point x is in the interior of the ellipse which passes through t 

 and has the points zero and unity for foci, 



can be expanded in the series of hypergeometric polynomials 



F/n+l, 



\ 



+ ...adinf. .......... ........ (30). 



If a; is outside the ellipse, the series is divergent. 



If x is on the ellipse, and t x is not zero, the sum of n terms of the series, when n 

 is large, oscillates in general between two finite limits. Hence when x is on the 

 ellipse the expansion fails. 



15. Multiply both sides of the last equation by <f>(t)dt and integrate round a 

 simple contour enclosing the points t = and t = 1 but no singularity of <j>(t). 



Theorem VII. Let <(z) be any analytic function which is regular at all points in 

 the interior of an ellipse C, whose foci are at the points 



z = and z = 1. 



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

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



Let p and y be any quantities, real or complex, subject only to the conditions that 

 the real parts of y andj> y+1 are positive. 



Then <(z) can be expanded in the infinite series of polynomials 



y, z) 



,-y,z)+ ..... (31), 



where 



, y, 0*W* 



