314 REPORT — 1873. 



Section 5. — These priuciples are then applied to the multiplication of 

 hyperelliptic functions. The following theorem is given without demonstra- 

 tion, ^3 3 (v, iv) being the same as before : 



'^h<Pz,z('"+"h' w + h'JP'i'-^)= 



1 



n—l »— 1 



2^ 2yA^_ye2^''+2y«'03, 3 (nv+%a^+(i log^p + 2yA, nw+'$\+2fiA+ylog^ q,p'\ q"An). 







where A™ is a constant analogous to Q,^ in the last section. 



To prove this formula we proceed as follows : the notation and assump- 

 tions will be understood by referring to Rosenhain, p. 400. To prevent con- 

 fusion, we write p for Eosenhain's n. 



nj^3,3(^ + V^'^ + ^A) = ^^ 

 1 



J,m^^+m^^+m^^ + +vi/)log^p+{n^^+n^'' + V)log,5. 



4:{m^n^+tti^n^+m.^n^ + Mp»p)A 



^2{mi+m.^+m^ + mp)v+2{ni+n.^+n^ + Wp)^^ 



2(^1^1 +«»i«2 4- m(,ap)+2{nib^+n.,b.^+ 4-«p''jp)_ 



Let 



"^=;"a+''^'' ^^^ ^^^^ Fi+^2+/^3+ ■ • • • +/'p'='''' 



so that 



n\ + m^+ .... +m =P + px, 



n 



h=^'h + y' ^^^ ^^^° »'i + »'2+ +*'p = 7> 



so that 

 Then 





n 



,^+n,/+....»; = S.,=+2yy+p.V% 



m^n^ + 7n^n^+ +mn^=I,lj^v^ + liy + y.v+pxy. 



Hence, collecting these resiilts, and resuming the (n) 



n 



1 



_ 2SA e2/3y+2yw ^ ^x" .n log p+4An.ri/+f/'' . n log^ ([ 



g2^(|3 log^^4-27A+2«^+My)e2^(2/3A+/3 log j+^Zb^+nw) 



= SA^_^e2/3i'+2yw . ^^_ 3(ni; + /3 logj> + 2yA + Sfl;i, m(; + 2/3A-|-/3 log^ 9 + 2ij,2>", <?", Ah) 

 where A^^ is a constant to bo determined (see p. 404). 



