514 ALGEBRA AND ANALYSIS 



defined by differential equations, the connection of this surface plays 

 an important role; this happens for geodesic lines. 



The notion of connexity, moreover, presented itself long ago in 

 analysis, when the study of electric currents and magnetism led 

 to non-uniform potentials; in a more general manner certain multi- 

 form integrals of some partial differential equations are met in 

 difficult theories, such as that of diffraction, and varied researches 

 must continue in this direction. 



From a different point of view, I must yet recall the relations of 

 algebraic analysis with geometry, which manifest themselves so 

 elegantly in the theory of groups of finite order. 



A regular polyhedron, say an icosahedron, is on the one hand the 

 solid that all the world knows; it is also, for the analyst, a group of 

 finite order, corresponding to the divers ways of making the poly- 

 hedron coincide with itself. 



The investigation of all the types of groups of motion of finite 

 order interests not alone the geometers, but also the crystallo- 

 graphers; it goes back essentially to the study of groups of ternary 

 linear substitutions of determinant +1, and leads to the thirty- 

 two systems of symmetry of the crystallographers for the particular 

 complex. 



The grouping in systems of polyhedra corresponding so as to fill 

 space exhausts all the possibilities in the investigation of the struc- 

 ture of crystals. 



Since the epoch when the notion of group was introduced into 

 algebra by Galois, it has taken, in divers ways, considerable devel- 

 opment, so that to-day it is met in all parts of mathematics. In the 

 applications, it appears to us above all as an admirable instrument 

 of classification. Whether it is a question of substitution groups 

 or of Sophus Lie's transformation groups, whether it is a question 

 of algebraic equations or of differential equations, this comprehen- 

 sive doctrine permits explanation of the degree of difficulty of the 

 problems treated and teaches how to utilize the special circumstances 

 which present themselves; thus it should interest as well mechanics 

 and mathematical physics as pure analysis. 



The degree of development of mechanics and physics has per- 

 mitted giving to almost all their theories a mathematical form; 

 certain hypotheses, the knowledge of elementary laws, have led 

 to differential relations which constitute the last form under which 

 these theories settle down, at least for a time. These latter have 

 seen little by little their field enlarge with the principles of thermo- 

 dynamics; to-day chemistry tends to take in its turn a mathemat- 

 ical form. 



I will take as witness of it only the celebrated memoir of Gibbs 

 on the equilibrium of chemical systems, so analytic in character, 



