Editing Gyrocompass (section)

=== The general and physically relevant case ===
Let us suppose now that <math>\sin\delta\neq0</math> and that <math>\alpha\approx0</math>, that is the axis of the gyroscope is approximately along the north-south line, and let us find the parameter space (if it exists) for which the system admits stable small oscillations about this same line. If this situation occurs, the gyroscope will always be approximately aligned along the north-south line, giving direction. In this case we find
<math display="block">\begin{align}
L_{x}&\approx I_{1} \left (\dot{\psi}-\Omega\sin\delta \right )\\
I_{2}\ddot{\alpha}&\approx \left (L_{x}\Omega\sin\delta+I_{2} \Omega^{2}\sin^{2}\delta \right) \alpha
\end{align}</math>

Consider the case that
<math display="block">L_{x}<0,</math>
and, further, we allow for fast gyro-rotations, that is
<math display="block">\left |\dot{\psi} \right |\gg\Omega.</math>

Therefore, for fast spinning rotations, <math>L_x<0</math> implies <math>\dot\psi<0.</math> In this case, the equations of motion further simplify to
<math display="block">\begin{align}
L_{x}              &\approx -I_{1} \left |\dot{\psi} \right | \approx \mathrm{constant}\\
I_{2}\ddot{\alpha} &\approx -I_{1} \left |\dot{\psi} \right |\Omega \sin\delta\alpha
\end{align}</math>

Therefore we find small oscillations about the north-south line, as <math>\alpha\approx A\sin(\tilde\omega t+B)</math>, where the angular velocity of this harmonic motion of the axis of symmetry of the gyrocompass about the north-south line is given by
<math display="block">\tilde\omega=\sqrt{\frac{I_{1}\sin\delta}{I_{2}}}\sqrt{\left |\dot{\psi} \right |\Omega},</math>
which corresponds to a period for the oscillations given by
<math display="block">T=\frac{2\pi}{\sqrt{\left |\dot{\psi} \right |\Omega}}\sqrt{\frac{I_{2}}{I_{1}\sin\delta}}.</math>

Therefore <math>\tilde\omega</math> is proportional to the geometric mean of the Earth and spinning angular velocities. In order to have small oscillations we have required <math>\dot{\psi}<0</math>, so that the North is located along the right-hand-rule direction of the spinning axis, that is along the negative direction of the <math>X_7</math>-axis, the axis of symmetry. As a side result, on measuring <math>T</math> (and knowing <math>\dot{\psi}</math>), one can deduce the local co-latitude <math>\delta.</math>