ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 337 



In the theorem of last section, let 



this is equivalent to assuming s=0, »i=l, \ = ji, 



al=al =....al =0, 



n 



I 



Hence ^ _^ _ n — 1 



1 a — g— , 



and the theorem becomes 



2^(^«i+^ — 2^ • • ''''+ ^~"2]r)^^^*^''"'- -""p' "^'-i- •'"^^p- p)- 



We shall apply this to prove the theorem for the transformation of the Abelian 

 integrals of the first order given on page 32. 



Put }t=3, p = 2; take n^n^ successively 0, 1, 2. 



Then 2^=0(Wj-^, 'i(o.-i) + t>(-u{u^-i) + <p{u, + ^, m,-|), 



+ 0K-3' W2 + 3) + K"i' "2+3) + 0("i + 3' «.+ 3)- 



= e(M,-|, ^«,-i)9KMJ0(«, + §, M, + |) ....... (1) 



+ e(u^u,-i)d{u^ + i, rg0(w, + |, «,+ f) (2) 



+e(u^+l M,-^)0K+|, «,)flK+|, tt,+ ^) (3) 



+ 0(«i-i «2)9(w., M. + ^)«K + i w. + l) (4) 



+(i(u^u;)e(u^+i, n^+ 1)6(11^+1 w^+i) (5) 



+ 0(^ + i «.)9K + f. w.+ i)fl(«,+|-, «3 + 5) • • • ' • • (6) 



+0K-i ".+i)0K, '*.+i)eK+i, "3+f) (7) 



+0(«,, w,+^)0(",+|, ^*a+i)eK+f. ^+f) (8) 



+ 0(«, + i, w. + ^)9K + |, ^*.+i)0(^+|, «.+|) (9) 



We see that lines (159), (267), (348) are identical; and the theorem of 

 last section therefore becomes 



= C0(3k„ 37f,, 3r,,,, 3r,,„ 3r,.,). 

 1S73. z 



