. = ON THE RECENT PROGRESS OF ANALYSIS. 71 
In his own Journal (v. 34 and 441), M. Liouville has since shown that 
elliptic integrals of the first and second kinds, considered as functions of the 
modulus, cannot be expressed in finite terms. 
95. In the eighteenth volume of the ‘Comptes Rendus’ (Liouville’s Journal, 
_ ix. 353), we find in a communication from M. Hermite, of which we shall 
shortly have occasion to speak more fully, a remarkable demonstration of 
Jacobi’s theorem. It is stated for the case of the first real transformation, 
but might of course be rendered general. This demonstration depends es- 
sentially on the principle already mentioned (p. 59), that any rational func- 
~ tion of a root of an algebraical equation which has the same value for every 
root of the equation is rationally expressible in the coefficients. The equa- 
tion to which this principle is applied is that to which we have so often re- 
ferred, viz. y = = considered as an equation to determine 2 in terms of y, 
and by means of it, M. Hermite shows at once that a certain rational func- 
tion of z is also a rational function of y, the form of which is subsequently 
determined. 
M. Hermite goes on to prove other theorems relating to elliptic 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 com- 
munication 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 va- 
lues of it, infinite; that from hence the known properties of elliptic func- 
tions are easily deduced ; and that by means of this principle he had suc- 
ceeded 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 appeared. 
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 contains a very ingenious proof of the fun- 
_ damental formula for the addition of two functions, derived from the differ- 
_ ential equation of the second order; which each function must satisfy. 
___ Other contributions to the theory of elliptic functions might be mentioned ; 
x 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 assumed, in consequence of the discoveries of 
_ Abel and M. Jacobi, is that which it will probably always retain, however 
_ our knowledge of particular parts of it may increase. What has since been 
effected relates for the most part to matters of detail, of which, howevér im- 
_ portant they may be, it is difficult or impossible to give an intelligible ac- 
- count. 
_ 26. It does not fall within the design of this report to consider the various 
_ applications which have been made of the theory of elliptic functions ; but 
_ Ishall briefly mention some of the geometrical interpretations, if the expres- 
_ sion may so be used, which mathematicians have given to the analytical re- 
sults of the theory. 
The lemniscate has, as is well known, the property that its arcs may be 
_ represented by an elliptic integral of the first kind, the modulus of which is 
si 
___ * M. Liouville has mentioned that M. Hermite had demonstrated the formule in question 
_ inadifferent manner. ¢ 
