328 Mr Olaisher, On theorems in elementary trigonometry. [Feb. 9, 



The theorems are in effect due to Professor H. J. S. Smith, 

 being readily deducible from the first four of the group of eleven 

 formulge for the multiplication of four Abelian functions given in 

 his paper " Note on the formula for the multiplication of four 

 -theta functions " {Proceedings of the London Mathematical Society, 

 Vol. X., pp. 91 — 100). These four formulae are 



(i) 2A1^ (a) Al^ (6) Al^ (c) Al^ (c?) 



= K\{a)K\{h')A\^[c')X\{d')+ p Al,(a')Al,(6')Al,(c')Al,(cZ') 



+ i,Al„(a') Al,(6')Al„(c') Al/cZ') - ^Al3(a') Al3(&0Al3(c')Al3(c^'). 



(ii) 2Al,(«)Al,(6)Al,(c)Al,(c?) 

 = Al,(a')Al(6')Al,(c')Al,(cr)+ ^ Al^(a')Al^(6')Al^(c')AlX<^') 



+ ^,Al3(aO Al3(&') Al3(c') AlsC^n " |? Al,(a')Al„(6') Al„(c')Al,(cZ'). 

 (iii) 2Al„(a)Al,(Z.)Al,(c)Al,(rZ) 



= Al„(a')Al„(6')Al„(c')Al,(cZ') + p Al3(a')Al3(6') Al3(c')Al3(cZ') 



+ ^^Al^(a)Al,(6')Al,(cOAlX^') - p Al,(a')A;(6')Al,(c')Al,((^'). 



(iv) 2Al3(a)Al3(6)Al3(c)Al3(c/) 



= Al3(aOAl3(6')Al3(c')Al3(c?')+ k" X\{a:)k\{h')A\{c')K\{d') 



+ /^^ Al, {a') K\ ih') X\ (c') Al, {d') - Fk"A\{a') Al,(6') Al,(c') Al^(cZO; 



and, in virtue of the equations, 



_ Al, jx) _ A\ {x) _ AI3 {x) 



Al„(a;)' Al„(^)' Al,(^)' 



these may be written 



(i)' k'h'^Ii{Bn a) = iM{Iifk"Il{sn a) +k''Il{cna')+k"-Tl{dna')}, 



(ii)' F U {en a) = lM{k^U {en a') +k'k"'U{sna)+U(dn a) -k"], 



(iii)' k" = I M {k"+ n (dn a) +k'k"Il{sn a') -FH (en a')], 



(iv)' -n.{dna)=^M[n{dna')+k"+k'Il(cna')-k'k"U{sna)], 



where 



A]„(a')Al„(5')Al„(c')Al,(^') . . 



Al„(a)Al„(6)Al„(c)Al„(tZ) ' 



