338 REPORT— 1873. 



From this Kdnigsberger deduces the well-known formulae for the transfor- 

 mation of elliptic functions of the third degree. 



Section 7. — This section opens with the following theorem 



(where 2 applies to ^^ . . . . pp, which are either or 1) : — 



B(v^....Vp, r,,i.. ..rp.p) = 20(2v^ 2vp, |^,j....|^^, 4r,,j. . . .4rp,p). 



Now 



%ei2v,. . . .2v„ if.,....ifi„ 4r,,....4rp.p) 



0(2^ + 2/i.r,,i + 2A'2r,,2 + 2/'3^i.3+ • • ■ • 

 2a;,-H2^^r,,,-f-2/.,r,,, + 2/i3r,,3. . . . 4r,. ^ . 4r,. , . . . .) 



2e-''i(2''i+2/^i^i. i+2/'2ri. 2+ • • • • +2vir,, ,+2,',r,, 2+ ■ ■ • ■ )^«. . . . 

 ^2i'^(2fj,+2/tir5, i+2/ijrj, 3+ . . . . -|-2r,r2, 1 4-2v2rj, 3+ . . . . )7ri_ 

 ^ ^^2v.(2t^,+2r,ri, , +2r,ri, ,+ .... ),re 



^2v,(2t>,-|-2,.ir, i+2i',r,. .-f- .... )'^' ... . 

 (where ;Uj = 0, f^^—O), 

 + .g(2t'i+r,,i + r,,, + ri,3) _ g(2y.-|-r,, i+r,, 2+75, ,+ . . . .)t«^ 



2g2v,(2t'i+2r,, ,+2r,, ,4- . . . . +2r.r,, , +2v,r„ ,+ . . . .)«^ 



^2v,(2«,-!-2r„ ,+2„ ,-|- . . . . +2,-ir,, ,+2v,r„ ,+ ....)« 

 (where ^^ = 1, //,=!), 



^_^(2,.,+2r„,-f-....)7r»^ 



Se2"i(2''i+2n, ,+ .... +2.',r,, , +2^,r,, , + .... )7rJ^ 

 ^2va(2i),+2r„ ,+ .... +2v,t^. , +2v,r„ ,+ . . . . )7ri. . . . 

 (where /Hj = 0, fi^=l), 



=x £»(v,v,.. ..Vp). 



The reader will see this if he will consider the following equivalences : — 



4''i'-i.a + 4''i>'jrj,, + 4r^r,,j-}-4f,i.^r,,,-t-r,,,-|-r,,i 



= (2r, + l)(2.,-hlK,+ (2.,-Hl)(2,.,+ l)r,.,; 

 also 



4r,r,„-F4v,v,r,.,-h4r,v,r,,, = 2,.,(2r,-flV,.,4-(2v,-Fl)2v,r,,,. 



