ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 103 



Putting >'= Wx, /i=* VX-, the general problem of modular equations, which 

 wo have to solve, is the following : — 



To determine the relation between \ and I; or fi and >', so that 



M being a constant miiltij)lier, and ?/ and x connected by the relation 

 a.v+l3x'' + yx'+ +a>a;2'»+' 



We easily derive the following theorem from the ' Fundamenta Nova : ' — 

 1 — I- sin am nu= 



u ^ . /u 4iA'\ ^ ^ . (u , 4(n-l>-A' \ 

 l-Xsinam^ . 1-Xsinmn(^^+ —). . ■ . 1-Xsin«m(^.^+ ^^^ j _ 



4^A' , 8/ A' . 2(n-l)/A' 

 A=«m-^. A^fl)H-jj- . . . . A=flm 



and substituting in this the values of the factors in the numerator derived 

 from the equation, 



4mK + 4m'?'K' 



1— Xsin«m( 



/».4>n'/A'\ n„.|l-X-sinrn.(^n+ ^^ -J j 



AM"^ 7i / f Ik ttK^ ^ 2(n-l)K1 "'■ 



^ J Aflm- — A am . . . . A am— ^ ^ 



)i n n J 



transforming by the formulas of page 37 of the ' Fundamenta Nova,' and 

 determining the constant multiplier by putting n=o in both sides of the 

 equation, we have 



\ — Tc sin am (nu) = 



-^ ^ j 1— 7c Bin a in u sin am ^l Y 



(l-Jc sin am ti) n,„' n,„ 4„iK + 4wi'?K' — ' 



-•Lzl ° l_Fsin^r( 



TT- J. — "• "^"^ ..Hi ttsm"a»i 



2 n 



2 



where, however, when »i=0, n,„' must be substituted for D 



n -l 

 2 



From this Sohnke deduces that symmetrical functions of the quantities 



4wK + AmUK! 



— - — '■ , when m and m' have the values just assigned, are 



sm coam 



n 



rational and entire functions of Jc. 

 Section 2. — We know that 



j/=^"{sin coam 4w sin coam 8o> sin coam 2(n — l)w} ; 



and it appears, from the ' Fundamenta Nova,' that if we put in this equation 

 successively the (h+1) values, 



_K iK' K + iK' K + 0^-l>^K' 



7i' 71 ' n ' ' n 



