ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 



351 



Transcendenten," in the tiftietli volume of the same Journal. These me- 

 thods have been admirably harmonized and developed by ScheUbach in his 

 ' Lehre von den elliptischen Integralen und den Thetafunctionen,' Berlin, 

 1864. I proceed to lay a summary of the results thus obtained before the 

 reader, accompanied by such remarks as may obviate difficulties. I commence 

 by writing in full the notation to be employed. 



d(.v)^l—2c/ cos 2x+2q' cos 4.r— 2g'' cos Gx+ . .., 



d^v) = 2^ I siQ X —2ql sin Sx + 2^ ^ sin 5a' — . . . , 



0,(.r) = 2rj 4 cos X + 2rfi cos 3.r + 2qT cos 5x-\- . .. , 



d.J^x) =l-\-2q cos 2x + 2q ' cos 4.r + 2q '■' cos 6.(- + 



If 5=e->', vfe shall sometimes use the following abbreviations for the 

 four series, d{x, v), Q^(x, v), d.,(x, v), 6.^{x, r). Let also 



/(.r) = 



Then 



sm am . 



(\x 

 dx ' 



2Kx 



r/x-. 



0£ 



ex' 



Ox ' 



A am 



2K,r 



1 ,.. s 2Tv.r J k' 

 = -r^Ax) , COS am = -I_i/ar, 



The periods of these functions are given by SchcUbach (section 22, p. 34). 

 By direct multiplication we find that 



«3^«3y=0.(^+y. 2r)d.Xx-i/, 2,0 + 9,(0; + //, 2y)d,ix-7/, 2y), 

 e xQ y=dXx + 7j, 2.)Qlx-ij, 2^>)-B,{x+y, 2v)Q.ix-y, 2y), 

 ^^^.y = U^+y,'2r)Q.Jix-y, 2v)-d,{x+ij,2v)Blx-tj,2y), 



e,xd.J/ = 6,(x + l/, 2y)e,(x-y,2y) + e,(.T + 1/, 2r)d,(x-y,2,). 



I need hardly point out the close analogy between these and the ordinary 

 trigonometrical formula;. 



From these formula; arc easily deduced the following : — 



eAK^+y)> h'WiK^-yl h'}=^'^'0.!/+(^^^.i/ 

 9 Hi^+y), h'K{ii-''-i/\ h'}=dxdy+e,xd^i/ 



0i{2(^+iO.i»'}MK-^'-3/). h'} = d,xey+d ae^y 



HKx^y),h-}^ {\{^-y), \v]^OxQy-Q^d^ 



2d (x+y, 2r)d (x-y, 2r) =9 .rd,y + d,xdy . 



2D,{x+y, 2v)e,(x-y, 2r) =6 xd,y-Q,xQy . 



2Q.J^x+y,2,')d.lx-y,2r) =e,xd,y-d xdy . 



2e^{x+y, 2v)dlx-y, 2v) =d,^d,y + d xBy . 



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