We examine the Boyarsky principle for generalized hypergeometric functions. This involves understanding the integral representations of such functions to the point of being able to show that the Frobenius matrix varies analytically with the multiplicative parameters (e.g. a, b, с in the case of the Gauss 2^i(fl> b, c, A). Following a suggestion of G. Laumon (letter of 25 April 1983) we use the Laplace transform to develop a theory broad enough to encompass the four hypergeometric functions of Appell (Appell and Kampe de Feriet 1925), their Lauricella generalizations and *i%_i (Erdelyi et al. 1953). Even in the case of 2F\ the present work represents a significant improvement over our previous study (Dwork 1983) of the Boyarksy principle. From the point of view of cohomology associated with hypergeometric functions our work represents an improvement over that of Aomoto (1975, 1977) who seems to have restricted his attention to cyclic coverings of P"1_1 ramified at a set of non-singular hypersurfaces with normal crossings (i.e. the intersection of each subset is a non-singular complete intersection). The condition is satisfied by 2F\, Fi of Appel and the Lauricella FD but not by the remaining hypergeometric functions listed above. In our theory we consider coverings ramified at
