ON THE RECENT PROGRESS OF ANALYSIS. 295 



essentially on the principle already mentioned (p. 276), that any 

 rational function of a root of an algebraical equation which has 

 the same value for every root of the equation is rationally ex- 

 pressible in the coefficients. The equation to which this princi- 

 ple is applied is that to which we have so often referred, viz. 



y = -y , considered as an equation to determine x in terms of ?/, 



and by means of it, M. Hermite shows at once that a certain 

 rational function of x is also a rational function of ?/, the form of 

 which is subsequently determined. 



M. Hermite goes on to prove other theorems relating to ellip- 

 tic functions. 



As elliptic functions are doubly periodic, we may determine 

 certain of their properties by considering to what conditions 

 doubly periodic functions must be subject. This view is 

 mentioned by M. Liouville in a verbal communication to the 

 Institute (Comptes Rendus, t. xix.). He states that he had 

 found that a doubly periodic function which is not an absolute 

 constant and has but one value for each value of its variable 

 must be, for certain values of it, infinite ; that from hence the 

 known properties of elliptic functions are easily deduced; and 

 that by means of this principle he had succeeded in proving the 

 expressions of the roots of the equation for the division of an 

 elliptic integral of the first kind, which M. Jacobi had given 

 without demonstration in Crelle's Journal*. I am not aware 

 that any development of M. Liouville's view has as yet ap- 

 peared. 



In the recent numbers of Crelle's Journal there are many 

 papers by M. Eisenstein on different points in the theory of 

 elliptic functions. Among these I may mention one which con- 

 tains a very ingenious proof of the fundamental formula for 

 the addition of two functions, derived from the differential equa- 

 tion of the second order, which each function must satisfy. 



Other contributions to the theory of elliptic functions might 

 be mentioned ; some of these, not here noticed, are referred to in 

 the index which will be found at the end of this report. But in 

 general it may be remarked that the form which the subject has 



* M. Liouville has mentioned that M. Hermite had demonstrated the formulae 

 in question in a different manner. 



