JULES HENRI POINCARfi. v 



These studies led directly to his discoveries in the field of auto- 

 morphic functions where Poincare achieved his first great celebrity. 

 Like most first-class things in modern mathematics, it is impossible 

 to describe these functions briefly in a non-technical discussion.^ 

 We must be content to characterize them as the nearest lying and 

 most beautiful generalizations of the trigonometric and elliptic func- 

 tions. Poincare deserves to be classed as one of the founders of 

 this branch of mathematics, for to him are due some of the main 

 outlines of the theory and the main existence theorems. Poincare's 

 last important contribution was his memoir on the zeta-fuchsian 

 functions in the Acta Mathematica for 1884. Since then this work 

 has been carried forward chiefly by Klein and his students. 



This series of contributions to function theory was followed in 

 1885 by his epoch-making memoir on the figure of equilibrium of a 

 rotating fluid mass. In this work he not only solved the problem of 

 stability for the previously known figures of equilibrium, the ellip- 

 soids of Maclaurin and of Jacobi, but he also discovered a whole 

 class of new figures of equilibrium. This work is important not 

 only on account of the particular new figures (the pear-shaped fig- 

 ures) which it put in evidence, but also on account of its method. 

 I need mention only the theorem on the exchange of equilibrium. 



In 1890 he made a still greater contribution to mathematical 

 astronomy in his memoir " On the Problem of Three Bodies and the 

 Equations of Dynamics." Here he brought into existence a general 

 theory of periodic orbits and disproved the existence of further new 

 integrals which are analytic functions of the masses. These matters 

 and many others, such as the integral invariants and asymptotic solu- 

 tions, are the subject of his three volumes (1892, 1893 and 1899) 

 on " Les Methodes Nouvelles de la Mechanique Celeste." Later on 

 he published three more volumes entitled " Legons de Mechanique 

 Celeste " in which he developed some of the classical theories from 

 new points of view. 



The problems of celestial mechanics continued tb occupy the 

 mind of Poincare till the end of his life. In his last paper, which 



^ This difficulty must indeed be my excuse for the summary way in which 

 I shall have to refer to the rest of Poincare's work. 



