ON THE RECENT PROGRESS OF ANALYSIS. 83 
mature of the inverse functions which are to be introduced. The number 
of these functions will of course vary in different cases, just as in M. Jacobi’s 
less general theory. Let us suppose this number to be denoted by y, then 
each function will involve y variables. And if each of these variables be 
replaced by the sum of two new variables, then all the functions are given 
as the roots of an equation of the yth degree, whose coefficients are rational 
in terms of the corresponding functions of each of the new variables and of 
certain known algebraical functions. From hence is derived the theory of 
the periodicity of these functions. 
After some other remarks on the theory of the higher transcendents, M. 
Hermite states that the method of division of which he made use in the 
_ problem of the division of Abelian integrals extends also to the new trans- 
cendents now considered, but that in the theory of transformation he had 
_ not as yet been successful. The greater part of the remainder of this re- 
markable communication relates to elliptic functions, and has been already 
: noticed. The remark just mentioned as having been made by M. Jacobi for 
the functions which are inverse to the Abelian integrals, extends, M. Hermite 
: observes, to the functions which he considers. 
In conclusion, M. Hermite remarks that the method of differentiation with 
respect to the modulus of which Legendre made so much use in the theory 
_ of elliptic functions, may be applied to all functions of the form 
Sf (ayaa, 
where y is given by the equation 
y'—-X=0. 
_ In concluding this report, it may be remarked that the subject of it is still 
_ incomplete, and that there is yet much to be done which we may hope it 
_ will not be found impossible to do. It is however difficult to predict the 
_ direction in which progress will hereafter be made. Yet I think we may 
_ reasonably suppose that the question of multiple periodicity, from the para- 
_ doxical aspect in which it has presented itself, and from its connexion with 
_ the general principles of the science of symbols, will sooner or later attract the 
_ attention of all philosophical analysts. M. Liouville’s idea of considering the 
 eonditions to which a doubly periodic function must as such be subject, can 
_ searcely be developed or extended to the higher transcendents without leading 
to results of great generality and interest. 
_ The detailed discussion of different classes of algebraical integrals, their 
transformations and reductions, form an endless subject of inquiry. But in 
this, as in other cases, the increasing extent of our knowledge will of itself 
tend to diminish the interest attached to the full development of particular 
portions of it; and with reference to analytical problems arising out of 
estions of physical science, the theory of the higher transcendents will it 
_is probable never become of so much importance as the theory of elliptic 
functions. We have occasion to make use of circular much more frequently 
than of elliptic functions, and similarly we shall, it may be presumed, have 
ss frequently to introduce the higher transcendents than elliptic functions. 
merical calculations of the values of the higher transcendents are therefore 
important than similar calculations in the case of elliptic functions*. 
* The Academy of Sciences has proposed as the subject of the great mathematical prize 
+ 1846 the following question :—“ Perfectionner dans quelque point’ essentiel la théorie 
des fonctions abéliennes ou plus généralement des transcendantes qui résultent de la con- 
‘sidération des intégrales de quantités algébriques.” The memoirs are to be sent in before 
the Ist of October. ‘ 
G2 
