ON ELLIPTIC AND HYPERBLUPTIC FUNCTIONS. 339 



Hence i{ z=Tfmx+-^tl\ , m= i., the equation (4) becomes 



(m'x' + mcx + ^!^'\ z" + m{2mx + c)z! -\-—z =0. 

 Therefore, differentiating {n—1) times, we have 



lmx' + mcx+—~\z -{■mn{2mx + c)z +kf!! -^^ "^ g =0; 



from this we have 



_ {2n—iym 



_2^___ 4(2»KV + C) 



n+- 



»Ha,- + cma'+— ^ („+i) 



z 



then 



2m^x + mc ' z!») 



an expression which of course leads to the development of (u) in a continued 

 fraction. 



Froni this Mahnsten obtains several resxilts. A very elegant consequence 

 of this investigation is the following theorem : — 



Let 



V I :iVl+.r^ J ' 



^jv_ 3.3. x^ 5.5. ■^- 7.7. x^ 9 . 9 . a;= 

 2+ 4+ 6+ 8+ 10+ ' 



which, with several others, will be found fully demonstrated in the paper. 



Part II. 



Section 1.— In entering on investigations relative to function 0,itis proper 

 to observe that two distinct notations are used to express the same four 

 series. It is now usual to write 



e x = l—2q cos 2.V+22* cos 4.^—229 cos 6x+ . . . 



i 9 25 



d^x= 2q sin .^'— 22isin3.r + 2gTsin5.v. , . 



d^x= 25^cos .r+22icos8.v + 22Tcos5.v 

 V'= 1 + % cos 2x + 2q^ cos 4^v + 2(f cos Go; + . . . 

 These four series are written in ' Fundamenta Jfova ' 



-^'^f,H25f,H??/..+;YeEf..+'Y 



TT ff \ 2/ TT \ 2/ 



The former notation is obviously more convenient ; and as uniformity is 

 most desirable, I shall usually adhere to it. It will, however, be occasionally 

 necessary to make use of the second, as it is employed in Crelle's Journal in 

 many papers which I shall have to bring before the reader. 



e- 



