Derive final theta-function formulas for the six direction cosines

Derive explicit, final-form expressions for the six direction cosines α, α′, α″, β, β′, β″ in the Kowalevski rigid body case with principal moments A = B = 2C and center-of-mass coordinate z0 = 0, expressing each as a rational function of quantities of the form θ_α(u1 + v1, u2 + v2)/θ(u1, u2) · e^{u3}, where θ_α denotes one of sixteen theta functions, u1, u2, u3 are entire linear functions of time, and v1, v2 are imaginary constants.

Background

The paper studies the rotation of a rigid body about a fixed point and identifies, in addition to the classical Euler–Poisson and Lagrange cases, a new integrable case discovered by Kowalevski characterized by A = B = 2C and z0 = 0. In this case she expresses the dynamical variables p, q, r, γ, γ′, γ″ using hyperelliptic and theta functions.

To complete the description of the motion, Kowalevski seeks explicit formulas for the six direction cosines α, α′, α″, β, β′, β″ that relate the body and space frames. She indicates that these quantities can be represented as rational functions of certain theta-function quotients with arguments linear in time but notes she has not yet derived their final explicit forms due to the complexity of the calculations. This leaves the task of fully writing down these formulas as an explicit unresolved item.