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



are the only possible singularities or branch-points of <(z) and its derivates, and sc 



t = \, . . . t = X, t = oo and t = z, 



are the only possible singularities or branch-points of the expression (19). 

 Hence, if we take a " Doppelumlauf " round two of the points 



t = AI, t == A 3 , . . . t = A,, 



we have a contour satisfying the condition. There are in general n1 independent 

 contours of this kind, which, together with <(z), which is given by a simple contour 

 round t = z, form a complete set of integrals of the equation (18). 



Some of the contours may become evanescent, either because the equation 



...+a n z" = 

 has equal roots, or because 



z = ,\,, ~ = A 2 , ...z = A B 



are not all singularities or branch-points of <(z). In that case the method does not 

 yield a complete set of integrals. In special cases a contour may reduce to a straight 

 path connecting two singularities. 



9. The theorem has been proved for the special case of the hypergeometric 

 equation in a posthumous paper of JAOOBI.* 



The result is of importance in the expansion of (t x)~ l in a series of hyper- 

 geometric functions. 



JACOBI proves further that if n is a positive integer 



(l-xY^} . (20), 



from which it follows after integration by parts that if the real parts of y+n and 

 p y+n+l are positive, 



z t 



- 



j. 



10. Theorem V. Consider the equations 



and 



z)v = o, 



* " Untersuchungon iiber die Differentialgleichungen der hypergeometrisehen Keihe," 'Crelle,' vol. 56. 

 Sec also JACOIH'S ' Gesammelte Werke,' vol. VI., pp. 184-202. 



