570 GEOMETRY 



to be essentially equivalent to one obtained by Castelnuovo in his 

 investigations on linear systems; and thus there was established 

 a connection between the so-called numerical and linear genera of a 

 surface, and the number of distinct double integrals. l 



A closely related set of investigations, originating with Clebsch's 

 theorems on intersections and Liouville's on confocal quadrics, may 

 be termed the "geometry of Abel's theorem." As later applications 

 we can merely mention Humbert's memoirs on certain metric pro- 

 perties of curves, and Lie's determination of surfaces of translation. 



Investigations in analysis have often suggested the introduc- 

 tion of new types of configurations into geometry. The field of alge- 

 braic surfaces is especially fruitful in this respect. Thus, while in the 

 case of curves (excluding the rational) there always exist integrals 

 everywhere finite, this holds for only a restricted class of surfaces; 

 their determination depends on the solution of a partial differential 

 equation which has been discussed in a few special cases. 



In addition to such relations between analysis and geometry, 

 important relations arise between various fields of geometry. Just 

 as an algebraic function of one variable is pictured by either a plane 

 curve or a Riemann surface (according as the independent and de- 

 pendent variables are taken to be real or complex), so an algebraic 

 function of two independent variables may be represented by either 

 a surface in ordinary space or a Riemannian four-dimensional mani- 

 fold in space of five dimensions. In the case of one variable, the 

 single invariant number (deficiency or genus p) which arises is 

 capable of definition in terms of the characteristics of the curve or 

 the connectivity of the Riemann surface. In passing to two variables, 

 however, it is necessary to consider several arithmetical invariants 

 just how many is an unsettled question. For the algebraic surface 

 we have, for instance, the geometric genus of Clebsch, the numerical 

 genus of Cayley, and the so-called second genus, each of which may 

 be regarded as a generalization, from a certain point of view, of the 

 single genus of a curve; all are invariant with respect to birational 

 transformation. 



The other geometric interpretation, by means of a Riemannian 

 manifold, has rendered -necessary the study of the analysis situs of 

 higher spaces. The connection of such a manifold is no longer ex- 

 pressed by a single number as in the case of an ordinary surface, but 

 by a set of two or more, the so-called numbers of Betti and Riemann. 

 The detailed theory of these connectivities, difficult and delicate 

 because it must be derived with little aid from the intuition, has been 

 made the subject of an extensive series of memoirs by Poincare". 



From the point of view of analysis, the chief interest in these 

 investigations is the fact that the connectivities are related to the 

 1 Comptes Rendus, February 22, 1904. 



