Appell Hypergeometric Functions

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series F1, F2, F3, F4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series F2, F3, F4 all have dimension 4, whereas that of the system corresponding to the series F1 has dimension 3. This chapter focuses on the F1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of S5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.

Some Elementary Projective Properties of the Four-Bar Linkage

The four-bar linkage defines a complete quadrilateral—a figure with extensive elementary projective properties. The undertaking is to search out and discover evidences or indicia of projective interrelationships in the geometry of the four-bar, such as concurrencies of lines, collineations of points, constancies of cross-ratios, harmonic divisions, tangencies of lines and curves, and others, toward the practical application of such properties in four-bar motion. Several of these have indeed been discovered and proved, each worth-while in its own right and each expressible as an instantaneous projective relationship between members of a simply-defined family of linkages. The most startling and applicable of the properties is the collineation of the centers of the inflection circles of the connecting-rod motions of the defined family.

An S-Configuration in Euclidean and Elliptic n-space

“The remarkable analogies which exist between the complete quadrilateral and the desmic system of points suggest that it may be possible to extend the properties considered above to spaces of higher dimensions”, remarks Prof. N. A. Court at the end of his paper (2). Here is an attempt in that direction in Euclidean as well as in elliptic 4-space, suggesting extensions in higher spaces. The corresponding figure, called an S-configuration, is discussed.