ON ELLIPTIC .VXD IIYPERELLIPTIC FUXCTIONS. 



where, replacing a, /?, y Ijv a + ^i'i, p + h'h y + l'''' '^^'^ ^'^"^'^ 

 t),\ e,iu Q,p ^ e,cc + d ^(\-oc)6(^^-cl.)e{p-a.) "1 

 dec 6(3 dy do • 9a9i(/J-«)^i(r-'") | 



e.(iS-<-g) e(\- /3)e(M-/3)9(p- /3) '^ 



e^ • 0/3 0i(a-/3) 0,(y-/3) 



6),(y + 3) e(\~y)eii.i-y)e(p-y) 

 OS ' dyd,{cc-y)6,{l3-y) 



Putting- p=y in (1), we have 



357 



+ 



+ 



(-^) 



d ,\dn _ e,(a.+h) B,(x-a) e,(V<-g) 0i(i3+ o) 



0,(\-/3) 6,(^-/3) . 

 «i/30,(a-/3) ' 



and (2) replacing X by a + |»'i, p by X, and y by a, we have 



X— 

 "6^ei(a-/3) 



0,xe,M _ g(«+g) e(x-g)9(;i-a) e(/3+^) e(x-/3) e(p-/3) 



0.S ■ eccd,l3—a M 



ea9/3 



where in each formula 2=X + /i — a— /3. 

 Putting 



X— a=rt, /3=&, fi—cc=:c, \+ji—j3=^d, 

 and then 



2s=rt + 5 + c + (?, 2(7=a + i + c— rf, 



these formulae become 



eadbdcdd =8,(7 d,(s—a) d,(s-h) d^(s-c) + ds d(<T—a) d(<T-h) d{a—c). 



From these it has been fully shown by Schellbach that the following for- 

 mulae may be derived, which were given by Jacobi under another shape in 

 the thirty-ninth volume of Crelle : — 



ho hx=hy h{x +y) +90 gxfyj\x-{-y), 



ho hy=hx h{x+y)+gofx gy f{x-\-y), 



ho gxfy=go hxj{x+y) -fx gy h{x-^y\ 



ho gyfx=go hyf{x+y)—gxfy h{x+y), 



ho gx/(x+y) =fx hy g(x +y) +go hxfy, 



fto gyfix+y)=hxfy g(x+y) +gofx hy, 



ho h(x+y)^hx hy—gofxfy g{x-\-y), 



hofx g{x + »/) =gx hyf{x -\-y) —gofy h{x +y), 



hofy g{x+y)= hx gy/(x +y) —gofx h (x +y) , 



hofx hyf{x+y) =go gy—gx g{x+y), 



ho hxfyf{x->ry)=gogx—gyg{x-\-y), 



ho gx hy g{x+y) =gogy hx h{x+y) —fx/(x+y), 



ho hx gyg{x +y) =go gx hy h{x +y) —fyf{x+y), 



hofxfy h {x\y) =gx gy —go g(x +y), 



ho gx gy h(x +y) =fxfy+go hx hy g{x -\-rj), 



ho hx hy h (,r +y) = ^ + go gx gy g{x + y). 



