ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 331 



Section 2. — We here supply the proof of the leading theorem given at 

 the commencement of this section : — 



^'. + 2|"^2, . + 1'4 +1^2 + i"'2 + 2?/2, . + 2|<r,, ,, 





= /^«^2t;^+9«;;+^;^+i«^r^^ l+i«l^^, 2+ • ■ )^\ 



c « 



(remembering that m'^^m'^+m'^ (mod. 2)) 



^g2i»;^(2i;^+,«^+2p^+m^+2(2?^+»^)r^ J;re_ 



^ ^-s(?^+K)(2'v+(?i+K)'-.,i+(?2+l"2)v2+- Ot*- 



(remembering that 23,Sn^r,,,=i%^2<r,,^+i2^^2:n^r„<,) 

 ( -1)2 (« w^ + 9 »«'")!ri -2 iw.''(»i^4-?«'')7ri „, 



-S(2.,+4<K2«'.+(?i+iOVl+(?2+*"2^^,2+--)Tt 



