870 



SCIENCE. 



[N. S. Vol. XX. No. 521. 



anaiysis situs an aid so fruitful as does the 

 theory of functions of one variable, owes 

 it, however, useful orientations. 



There is also another order of questions 

 where the geometry of situation intervenes ; 

 in the study of curves traced on a surface 

 and defined by differential equations, the 

 connection of this surface plays an im- 

 portant role; this happens for geodesic 

 lines. 



The notion of connexity, moreover, pre- 

 sented itself long ago in analysis, when the 

 study of electric currents and magnetism 

 led to non-uniform potentials; in a more 

 general manner certain multiform integrals 

 of some partial differential equations are 

 met in difficult theories, such as that of 

 dift'raction, and varied researches must 

 continue in this direction. 



From a different point of view, I must 

 yet recall the relations of algebraic an- 

 alysis with geometry, which manifest them- 

 selves so elegantly in the theory of groups 

 of finite order. 



A regular polyhedron, say an icosahe- 

 dron, 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 polyhedron 

 coincide with itself. 



The investigation of all the types of 

 groups of motion of finite order interests 

 not alone the geometers, but also the crys- 

 tallographers ; it goes back essentially to the 

 study of groups of ternary linear substitu- 

 tions of determinant -j- 1, and leads to the 

 thirty-two systems of symmetry of the 

 crystallographers for the particular com- 

 plex. 



The grouping in systems of polyhedra 

 corresponding so as to fill space exhausts 

 all the possibilities in the investigation of 

 the structure of crystals. 



Since the epoch when the notion of group 

 was introduced into algebra by Galois, it 



has taken, in divers ways, considerable de- 

 velopment, 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 equa- 

 tions or of differential equations this com- 

 prehensive doctrine permits explanation of 

 the degree of difficulty of the problems 

 treated and teaches how to utilize the 

 special circumstances which present them- 

 selves ; thus it should interest as well mech- 

 anics and mathematical physics as pure 

 analysis. 



The degree of development of mechanics 

 and physics has permitted giving to almost 

 all their theories a mathematical form^ 

 certain hypotheses, the knowledge of ele- 

 mentary laws, have led to differential rela- 

 tions 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 prin- 

 ciples of thermodynamics ; to-day chemistry 

 tends to take in its turn a mathematical 

 form. 



I will take as witness of it only the cele- 

 brated memoir of Gibbs on the equilibrium 

 of chemical systems, so analytic in char- 

 acter, and where it needed some effort on 

 the part of the chemists to recognize, under 

 theii" algebraic mantle, laws of high im- 

 portance. 



It seems that chemistry has to-day gotten 

 out of the premathematic period, by which 

 every science begins, and that a day must 

 come when will be systematized grand 

 theories, analogous to those of our present 

 mathematical physics, but far more vast, 

 and comprising^ the ensemble of physico- 

 chemic phenomena. 



It would be premature to ask if analysis 

 will find in their developments the source 



