ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 339 



To illustrate the series of equations next following, I observe : — 



Hence, substituting v^ — ^ iu the value of d(2v^ .... 2t'p, ^ja.j .... Ifip, 

 4r,,j. . . .4rp,p) just given, the expression becomes 



2(-lf20(2t>^....2t;p, iiu,....i/ip, 4r,.,....4rp,p). ... (A) 



From this series of equations values are deduced for 



0(2v,....4r„,....) and 0(2i»^. . . . iyu,. . . .4r,,,) 

 in terms of 



e(Vj....r,,i....)a. 



Putting p = 3 in the theorem at the commencement of section 3, and then 

 for Mj, Mj— 2, &c., an expression is found for 



%e{u^ + v^ ^tp + V ''1,1 '•p,p)a0K-3v, 3r,,i )a. 



a 



Modifying this by the equation for 0(2?;^.. -^.Ar^^ j), which we have 

 just mentioned, we have 



20K + ^-. • • V'p ' '■-.l- • • •'■p.p)a(«.-3^',. . • .3r,^,. . . .)a 

 a 



^ ft y 



Now we observe here that the index of ( — 1) in both cases is a series of 

 negative units, every one of which is multiplied by a quantity which is and 1 

 alternately, Hence, in taking the sum, the expression vanishes except for 

 y=^, and we have, when v^:=v^= .... =Vp=0, 



20(m^ . . . . Mp, r,^ 1 . . . . rp_ p)^0K . . . . Mp, 3r,^ 1 • • • • 3rp, p), 



a 



= 29(2«,....2m^; 3r,,,....3rp,p)„e(0....0, r,, , . . . . Tp, p)„. 



a 



From this we easily obtain, bearing in mind the method by which expres- 

 sion (A) was found, 



20(0. . . .0, 3r,, , . . .3rp,p)„^^0(O. . . .0, r,, ,. . ..r^,X^ 



a 



=2(-l)f >'0(O. . • -0, 3.,, ,. . . .3rp,p),0(O. . . .0, r,, ,r. . . .r^,^)^. 



a 



From this formula Konigsberger deduces three modular equations for 

 hyperelliptic functions of the first order. Since 3p — 3 is in this case 3, and 

 as this number is taken with one exception, the number of terras in the first 

 member of these equations is 2, the four terms in the second member corre- 

 spond to the values u„ v, ; v, — |, v^ ; v^, v^— | ; v^ — |, v, — 4. 



z2 



