ON ELLIPTIC AND HYPEEELLU'TIC TUNCTIONS. 



353 



then 



m:^,Mf(.-y))=% . y.»+y'«(..-rt+^f-y)» (£±gj 



00= 



0^- 0^ 0^ 



by (5), or 



90 7ioT{f{x+?/) +f(x-i/)} =fxgij hi/ . 

 In the same way may be derived the following :- 

 go ho T{f{x +y) -f{x -y)} =fy gx hx . 

 ffoT{g(x-7/)+g(x+y)}=ffXfft/ 



go 1{g{x-i/)-g(x+j/)} =fx hxfy hy 



(11) 



(12) 

 (13) 

 (14) 

 (15) 

 (16) 

 (17) 

 (18) 

 (19) 

 (20) 

 (21) 

 (22) 



From these formulae the ordinary expressions for the addition and subtrac- 

 tion of elliptic functions may of course be easily deduced. 

 We also find 



0/0 = 00 0,0 0,0, f'o = B.fl6,o, \ .. 



f,v= 60" gx hx, g'x = — d^d'/x hx, h'x = — O./rfx gx. J 



It is obvious that from the above the following may be immediately formed 

 by addition, &c. : — • 



holL{h{x—y)-]-h{x-\'y)} = hxhy 



hoT{h{x-y)-h{x+7/)}=fxgxfygy .... 



go TIJ(x +y)h{x —y) + f{x-y)h(x + ?/)} =fx hx gy 



go '^{f{3:-\-y)h{_x-7j)—f{x-y)h(x+y)}=fy hygx 



hoT {f(x+i/)g(x-i/) +f(x -y)g(x +y) } =fx gx hy 



ho T{f{x+y)g(x-y)-f(x-7j)g{x+y)}=fygy hx 



go ho T{g(x+yyi{x—y) +g(x-y)h(x+y)}=gxhx gy hy 



go hoT{g(x+i/)h(x—y) —g(oc-y)h{x^-y)] --=fx gxfy gy 



gohof(x+y) 



_faffyhy+f ygxhx 



.fx"—fy' 



^g^Jxyxhy^fygyhx ^g^.j^^,_ 



9X gy +Jxjy hx hy Jx gy hy -Jy gx hx 

 ho fx gy hx ■\-fy gx hy 



(23) 



go' hxhy+fxfygxgy 



gog{x+y)-- 



gx gy—fxfy hx hy 



hx"-h?f-l 



^-J^fy' 



_g^2fjxjixhj/—fyjfyhx^ 



g-eyy+f^fyh 



gx gy hx hy^fxfy 



_1 

 xhy I 



Jx gy hy -jy gx hx hx hy +fxfy gx gy 



ho h{x-\-y)= ^^^ y-f^fyy^yy —f^ fy+y^yyft^Jiy 



^-f^'Jr gxgy^fxjyhxhy 



^]^^-Jxgy l^x-fygx hy _ \-^gx-gy- 



fa yy hy —fy yj: hx hx hy -^rfxjy gx gy 



From these formulae we may deduce as follows -.^ 



J 



(24) 



(25) 



