144 Obituary Notices of Fellows deceased. 



Functions," which in the course of 160 pages gives an admirable out- 

 line of that theory. 



As is shown by the list of his papers, Hermite wrote on many topics 

 within the range of analysis : the subjects which recur most frequently 

 are the theory of numbers, invariants and covariants, definite integrals, 

 theory of functions, theory of equations, and elliptic functions. If 

 special mention may be made of advances that are due to him, and of 

 substantial discoveries achieved by him, instances can be selected from 

 each of those subjects. 



Thus, in the theory of numbers, he connected the use of continuous 

 variables with quadratic forms : and he introduced conjugate indeter- 

 minates into the discussion of those forms. Perhaps the most wonder- 

 ful of all his researches in this region was his proof (1873) of the 

 transcendence of e, the base of the exponential function a proof 

 which, duly modified, led Lindemann to the establishment of the 

 transcendence of TT, and so showed the quadrature of a circle to be 

 impossible. 



In the theory of invariants and covariants, where he was a fellow- 

 worker with Cayley and Sylvester, Hermite had an important share. 

 He was responsible for the law of reciprocity whereby, to every co- 

 variant of degree n in the coefficients of a quaritic of order m, there 

 corresponds a covariant of degree m in the coefficients of a quantic of 

 order n. He discovered the skew invariant of the quintic, which was 

 the first example of any skew invariant. He discovered the linear 

 covariants belonging to quantics of odd order greater than 3, and .he 

 applied them to obtain the typical expression of the quantic in which 

 the coefficients are invariants. He also invented the associated co- 

 variants of a quantic ; these constitute the simplest set of algebraically 

 complete systems as distinguished from systems that are linearly 

 complete. 



In the theory of functions, it is almost difficult to select representa- 

 tive instances from among his many contributions to that subject, in 

 which he may be regarded as the foremost of French writers since 

 Oauchy. Not the least important are the special advances he made 

 in the transformation of the double theta-functions and the associated 

 Abelian functions : his memoir has been the suggestive starting-point 

 for many other investigations. Anyone acquainted with the progress 

 of the subject in the last 30 or 40 years will recognise that Hermite 

 has given an entirely new significance to the use of definite integrals 

 in the theory of functions : it is enough merely to mention the de- 

 velopments of the properties of the gamma-function which have been 

 thus initiated. Indeed, he seems to have been the one mathematician 

 of the later half of the nineteenth century who could work easily with 

 definite integrals, almost recalling the fruitful activity of Euler in the 



