ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 



339 



is constant, or 



«^(«i + Kj....)o«^K-- -Oo 



is constant, applying Abel's theorem as before. Hence we see the truth of 

 equation (1). 



Section 2. — Hence it easily follows that 



aZ(M, + 2K,....)„=+«?K •-.•)„. 



the last formula of course holding good when /3 and a arc unequal. 

 Hence, if 



1 2m 



and 



f = /^o + /'i+ +/^2n' 



where fj^, n^. . ■ ■ are any whole numbers, then 



«7(«. + 2K,. . ..)^=(-lf^al(u^.. . .)„. 



Also let 

 and 



'^.=™iK^_i + m,K^2+....+w„K^_„, 



where m , m ... .m\, m\ are any whole numbers; then we find the two 

 following formulae : 



lit „ 



where, when a=Oj, m^ must be taken as zero, and 



aZK + 2.',i. . . .)„=(-ir'«-^-'-l + - ••"^^+1«ZK. . . .)., 



in which formula, when a.=2n, the multiplier of al(it^ . . . . )2„ is to be taken 



as unity. 



Section 3. — Let 



1 2 P« J«2,_i.^-« 



dx 



(1) 



P.r fZa,' 



n n Q(«2c-l) 



which last formula may be written thus : 



2.\ 2 



