Charles Her mite. 145 



same medium, but with an accuracy and a precision that were im- 

 possible in Euler's day. 



The theory of elliptic functions, on the Jacobian rather than on the 

 Weierstrassian basis, is selected by Picard as having been Hermite's 

 favourite study ; and the selection appears justified by the number of 

 his papers on the subject. To him is due the reduction of an elliptic 

 integral to its canonical form by means of the syzygy among the con- 

 comitants of a binary quartic. His investigations on modular functions 

 and modular equations are of the highest importance. It was Hermite 

 who discovered pseudo-periodic functions of the second kind, and 

 developed their properties. In a memoir that may be fairly described 

 .as classical,* he applied these functions to the integration of the 

 unspecialised form of Lame's differential equation ; and elliptic func- 

 tions generally were applied in that memoir to obtain the solution of 

 a number of physical problems. 



In the theory of equations, there are two significant contributions 

 which will be specially associated with his name. One of these is the 

 form which he gave to Tschirnhausen's transformation, with its accom- 

 panying property of securing invariance for the transformed equation ; 

 he also applied it to obtain invariantive criteria for the reality of the 

 roots of a quintic equation. The other of the contributions indicated 

 is the actual solution of the quintic equation by means of modular 

 functions ; in this result of supreme importance he was followed by 

 Kronecker and Brioschi. 



One quality of his papers deserve notice : it is the singular clear- 

 ness with which they are written. It is a commonplace to refer to 

 style as a characteristic of French writers ; but mathematical investi- 

 gations do not always lend themselves to clear and finished exposition. 

 Hermite's papers are remarkable for this quality. 



This brief sketch may be sufficient to give a slight indication of the 

 range of activity of Hermite's genius. His interest in mathematics 

 remained undiminished to the last. For many years he had been 

 regarded as the venerated chief among French mathematicians, sus- 

 taining the great traditions of the past and sympathetic with the 

 rising workers of his later days. 



A. E. F. 



* " Sur quelques applications des fonctions elliptiques " ; it appeared in the 

 ' Comptes Eendus,' vol. 85 (1877), vol. 86 (1878), vol. 89 (1879), vol. 90 (1880), 

 vol. 93 (1881), vol. 94 (1882) ; and the various parts were afterwards gathered into 

 a quarto volume of 146 pages (Paris, Gauthier-Villars, 1885). 



