334 REPORT— 1869. 



Report on Recent Progress in Elliptic and Hijperelliptic Functions. 

 By W. H. L. Russell, F.R.S. 



I SHALL never forget tlie evening when I first became acquainted -with Mr. 

 Ellis's report on the present state of Analysis published by this Association. 

 I felt like a traveller who, on entering an unknown country through dark 

 and narrow paths, suddenly arrives at an eminence from which he sees the 

 whole region spread out like a map before him, and perceives at a glance the 

 roads leading to the principal cities, and the most desirable mansions. I fol- 

 lowed this guidance in my reading, and, as I proceeded, became anxious to 

 attempt for others what Mr. Ellis had effected for me, and in consequence to 

 imdertake the present Report. 



I shall keep steadily in view three main objects. In the first place, I shall 

 endeavour to prove theorems enunciated by their authors without demonstra- 

 tion. Secondly, to explain passages which may present difficulties, and to give 

 such directions as may enable the reader to arrive with the least difficulty at 

 the most important parts of the different memoirs which will come under re- 

 view. And, thirdly, to give such a connected view of the whole, as will 

 enable anyone entering on the studjr of our present subject to know before- 

 hand the nature of the results which have been obtained. 



Elliptic functions will first be considered ; and I shall divide the subject 

 into four parts. 



(1) I shall consider recent researches in this branch which do not involve 

 the idea of periodicity. 



(2) Eeccnt investigations relative to function 0, and its allied series. 



(3) Modular equations, and some other researches of a similar description. 



(4) Some of the most important geometrical and physical applications of 

 eUiptic functions. 



Part I. 



Section 1. — It will be proper to commence by giving a list of the principal 

 algebraical integrals which can be reduced to elliptic functions. They are 

 taken from Schellbach's ' Lehre von den EUiptischen Integralen,' and from a 

 paper by Eothig in the fifty-sixth volume of Crelle's Journal. Along with 

 the integrals I shall indicate the transformations necessary for their re- 

 duction. 



J^ 



(l-enia^) . . . (1) 



V a„ -f- a^.^' + a.,x^ + a.jc'' -\- a^x* + «,a;* + a„«° 

 Let z—a;+-, I know not if it has ever been remarked that the same 



X 



, C cixn(x) 



substitution wiU enalne us to integrate I . , m loara- 



jVa+bx + cx' + bx^ + ax* 



rithms or circular arcs. 



Hence we can reduce to elliptic functions 



(2) 



J V «o + (i^x- + a.,.v* -f- a^x^ + «„, 



by putting x^=z, to which can be reduced 



