  The general analytic solution to the functional equation $$ \phi_1(x+y)= { {
\biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over {
\biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is
characterised. Up to the action of the symmetry group, this is described in
terms of Weierstrass elliptic functions. We illustrate our theory by applying
it to the classical addition theorems of the Jacobi elliptic functions and the
functional equations $$ \phi_1(x+y)=\phi_4(x)\phi_5(y)+\phi_4(y)\phi_5(x) $$
and \[ \Psi _1(x+y)=\Psi _2(x+y) \phi_2(x)\phi_3(y) +\Psi_3(x+y)
\phi_4(x)\phi_5(y). \]
