ON ELLIPTIC AND HYPEHELLIPTIC FUNCTIONS. 335 



To make this more clear I add the following proofs of some of these equi- 

 valences (see Table, p. 22) : — 



The other formulae in the Table may be proved in a similar manner. 



Section 4. — Konigsberger in this section gives the following theorem 

 (without demonstration) : — 



If 



0(«, . . . .u,)=d(m['hc^ + a[]\ .... <'Mp + a;", s^ . . . .s^') 



.... 0(m>, + at • . . . mi«p + «p, s(^> .... s<^)), 

 then 



2e-2r7{2r(/„+2«;,-(Sir^,, + . • • . +Spr,,,p)}5rj 



x4*> + 7;- J^^' + ''''^-+ • • +Sp-i.p)- •^V + '^'- J(Ap + S,rp„+ . .Sprp.p)) 

 = C0(mj..n«p, »T,,,..)Yp,p), 



where the summation with regard to the indices w, ... .rip extends from to 

 *' — 1, and r, A, S are given by the following equations : — 



,„(/)^ + .... + ^(^)^ = r, 



m(i>a(i) . . . . + m<^>a(^) = A„ 



m</'5(i^ + ....+ mW<^> = S,. 



I have worked out this theorem for hyperelliptic fimctions of the first 

 order; and it appears from this that the demonstration for hyperelliptic 

 functions docs not differ in principle from that for elliptic functions. I shall 

 therefore confine myself to elliptic functions, as the length for hyperelliptic 

 functions is extremely great. 



Putting then p^l, the theorem becomes 



2,-';(2r«+2«-S,r, 0-^^(,,+ '1 _ i(A, + S,r,. .)) = C0(rw,. ,rr, ,). 

 For \ = 1, this equation reduces itself to the following — 



or 



which leads at once to the equivalence 



S 2 i'(2wtM+— +vrj^j)Tri_.Q-^^v{2m^u+vm^Tj^,)7ri^ 

 Put in this equation p=v'm+fi, where fi is less than m. Then we have 



