280 ON THE RECENT PROGRESS OF ANALYSIS. 



can pass from one root to another along each horizontal row, 

 yet we cannot pass from row to row. 



Our table, however, has the remarkable property, that sup- 

 posing, as we may always do, 2n + 1 to be a prime number, all 

 the roots are rationally expressible in terms of any two not 

 lying in the same row. This depends on a property of the 

 function </>, which it is very easy to demonstrate, and it is inti- 

 mately connected with the relations which exist among the 

 terms of the same row. 



If, then, which is the case for an infinite variety of values 

 of the modulus, we can express any root rationally in terms of 



another of a different row, say in - - - , all the roots become 



Afl ~}~ i. 



rational in terms of <f> - -- - . Moreover, it appears that not 



only are the roots all expressible in one, but they are so in such 

 a manner that the functional dependencies among them fulfil 

 a certain simple condition, which, as Abel shows in a separate 

 memoir (Crelle, IV. p. 131 ; or Abel's Works, I. p. 114), renders 

 every equation, all whose roots are rationally expressible in 

 terms of one, algebraically soluble. 



To take the simplest case, the arc of the lemniscate may be 



f dx 

 represented by the integral I If <f> be the function in- 



verse to this integral, we have the simple relation between roots 



f TfV JL ., , . . . . 



of different rows, 9 - = tq> - - , &> being in this case 

 r 2n + 1 r 2n + 1 



equal to OT. 



To apply what has been said to the solution of the general 

 equation for determining <a in terms of < (2n + 1) a, it is suf- 

 ficient to remark that the transcendents introduced in consider- 

 ing the relations among the roots of this equation, are simply 



(f> - and , or at least may be algebraically ex- 



2,11 -j- 1 271 -f- 1 



pressed in terms of these two quantities. 



The remainder of the first memoir contains developments of 

 the functions <,/, and Fin doubly and. singly infinite continued 

 products and series. They are derived from the expressions 



of 0a, &c. in terms of </> - , &c., by supposing n to increase sine 



