  A stream of new theta relations is obtained. They follow from the general
Thomae formula, which is a new result giving expressions for theta derivatives
(the zero values of the lowest non-vanishing derivatives of theta functions
with singular half-period characteristics) in terms of branch points and the
period matrix of a hyperelliptic Riemann surface. The new theta relations
contain (i) linear relations on the vector space of first order theta
derivatives which are arranged in gradients, (ii) relations between second
order theta derivatives and symmetric bilinear forms on the vector space of the
gradients, (iii) relations between third order theta derivatives and symmetric
trilinear forms on the vector space of the gradients, and (iv) a conjecture
regarding higher order theta derivatives. It is shown how the Schottky identity
(in the hyperelliptic case) is derived from the obtained relations.
