ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 309 



In putting 



i-rr f , lit II , if III *T 

 we have 



v + '-l, .' + J, v"+l^, ."'-| £0TV,V',V",V"', 



d(:v)d{v')d{v")div"')-d,{v)d,{v')d^{v")d,{v"') 

 =0(i;jfl«)eK')0K")-0x(^)0i«)0iK")«i(^"');- • • (2) 



and if we substitute t^"' + iV in the place of v'" in these two equations, we 

 shall have : — 



e,(v)e,(vy^{v'')d,{v''')-d,(v)o,ivy,(v-y,iv''') 

 e{v)d(v')div")e{v"')+d,(v)e,(y')d,{v")e,iv"') 



= 93(^)03«)03(^")«3(^"')-0.(^)0.(^')0.(^")0.(^"')- • • (4) 



Section 2. — Putting, then, for a moment 



we have, adding (1) and (3), secondly subtracting (3) from (1), thirdly adding 

 (2) and (4), fourthly subtracting (2) from (4), 



2e(2>=0/«)+0/2>-0,-0/^>, 



from which 



0". + ea)2 + 0(2)2 + 0(3)2== 0^2 ^ 0)2 + 0^(2)2 ^ ^^(3)2^ ^^ 



{d,vd,v'dydy\^+{e^v6^v'6ye,v"'Y+{d^veydy'ey"y+ {ev6v'ev"ev"'y 



remains unchanged when v^, v\, v'\, v"\ are put for v, v , v" , v". 



This and four other formulae of a similar nature, obtained by augmenting 

 the arguments by semiperiods &c., are given by Eosenhain, and constitute 

 the starting-point from which he deduces the properties of the hyperelliptic 

 functions, as we shall soon see. See also a memoir by Professor Smith on 

 this subject in the 'Transactions' of the London Mathematical Society. 



Section 3. — Conceive now a function thus defined : 



00 

 03, ,{v, iy)=2,„p"'^e2'"''03(ty+2mA, q), 



-00 

 00 



= S„ q"-e^''%(v + 2nA , p), 



-00 



00 CO 



-oo-oo 



m^log p-^7i^log q-\-4tnnA+2niv-{-2nw 



This series is a function, doubly periodic (see Rosenhain, p. 389), of v and w 

 in the pairs of conjugate indices iw and and and in ; for we have 



