ox ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 333 



Q= 



+ ..,(20 + lK+^^ + 2(^,r,,,+ . .iu,r,p) + (^ + l)(V„,+ ..))+■•.. Ui, ■ 



from which Konigsberger's formula may immediately be derived, where, 

 however, the letter i must be exchanged in several places for the number 2, 

 for which it is plainly intended. 



Putting p = l, and multiplying the exponential partly into the function d 

 in u, and partly into the function d in v, and recaUing the definition of 



d(v^v^ .... t'p, nn^n^np) 

 given in the first section, we have at once 



e(u^ + t\. .«p + rpr,,,. .rp,p)0(H,-r',. .t'p-^'p. ^1,1- ■rp,p) 



= Se(2H, ..2»p, i^,..|/xp„ 2r,,..2.p,p)9(2t-,..2i;p, i//,..^/.p, 2r,,. .2rp,p). 



A formula is next deduced for 



d{u^ + v^ + w^. . .)d(u-v^. . . .). 

 We have moreover 



0(U, + Wj Mp + tf p, r,. , Tp, p)a0(l<i Upr,_ 1 rp,p)a 



= 29(2Mj + w;, 2?«p + Wp, l/ii . . . . |)Up, 2ri_ , rp, p)Q^, 



where Q^ is not connected with u. 



To prove this, we observe that, if we put Vj = in the last formulae, we 

 are able to show that 



%6(2u^ + w, + m1 + nj^^^+ . .2u^ + w, + 7n-+n^r,^^. ^H.- -if^p- .2r,, ,. .2rp,p)P(i>. 

 But 

 0(2M, + t., + m^ + n,r,,,. .|^,. .|;/p. .2r,, , . . 2rp,p) 



Combining these two expressions together, we see that the theorem is true. 



From this equation, by using 2'' values of (a) in succession, and elimi- 

 nating, we may obtain each of the 2'' values of 



d(2u^ + w^ 2«p + Wp, §/ii ifjp, 2r,,i. . ..2rp,p) 



corresponding to the 2'' values of /x^ .... jup in terms of a series of functions 

 of the form 



0(m Up, r,,i rp,p)a0(« + Wi Mp + Wp. r,,, rp.p)^, 



whence the formula above mentioned for 



e(u,+v^+iv^. . ..)e(n^-t\.. . .) 



