Generalisation of Jacobi's θ-Function Formulae

Jacobi's well known formulae on the multiplication of θ-functions can be generalised by making use of general orthogonal linear substitution. The generalised theorem runs as follows:-

If variables (l, m, n,...... p) and (l₁, m₁, n₁,..... p₁) are connected by means of the relations

(i){a₁₁l+a₁₂m+........a_1p P = a_1ll₁+a₂₁m₁+.......a_p1 

      _{a2ll+a22m+ ...,...a2p P = a12l1+a22m1+......ap2P1}

      _{...........................................................................}

      _{............................................................................}

     _{ap1l+ap2m+.........app P=a1pl1+a2pm1+......appP1}

_{where aik=-aki.}