314 ON THE RECENT PROGRESS OF ANALYSIS. 



where y is given by the equation 



y-z=o. 



In concluding this report, it may be remarked that the sub- 

 ject 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 sup- 

 pose 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 sym- 

 bols, will sooner or later attract the attention of all philosophical 

 analysts. M. Liouville's idea of considering the conditions to 

 which a doubly periodic function must as such be subject, can 

 scarcely 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 in- 

 tegrals, 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 in- 

 terest attached to the full development of particular portions of 

 it; and with reference to analytical problems arising out of 

 questions of physical science, the theory of the higher trans- 

 cendents 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 less frequently 

 to introduce the higher transcendents than elliptic functions. 

 Numerical calculations of the values of the higher transcendents 

 are therefore less important than similar calculations in the case 

 of elliptic functions*. 



The following index is intended to contain references to all 

 the papers in the first thirty-one volumes of Crelle's Journal, and 

 in the first ten volumes of Liouville's Journal, more or less con- 

 nected with the subject of this report, together with a consider- 

 able number of others. 



* The Academy of Sciences has proposed as the subject of the great mathe- 

 matical prize for 1846 the following question: " Perfection ner dans quelque point 

 essentiel la the'orie des fonctions abeliennes ou plus gdneralement des transcen- 

 dantes qui re'sultent de la consideration des integrates de quantite*s algdbriques." 

 The memoirs are to be sent in before the ist of October. 



