 Solution

Let $\hat{x}$ point to the equatorial meridian at zero longitude, $\hat{y}$ to the equator at $90^{\circ }$ E, and $\hat{z}$ to the north pole. Use a right-hand positive Euler angular velocity vector. Relative rotations add as vectors, not as geographic coordinates:ConsequentlyHere $\lambda ={\tan }^{-1}(12/5)$ is the longitude of the positive Euler pole, and its latitude $\varphi$ is zero. The antipodal pole with the opposite angular-velocity sign is an equivalent convention; mixing a pole reversal with an unchanged sign would give the wrong motion.