The transformation describes a latitudinal plus a longitudinal rotation: A rotation of the coordinate axes by an angle, , from the geographic pole along the Greenwich meridian (0 ^oE geographic). That is, a rotation about the y-axis in Figure B1(a). Plus a rotation by an angle, , in the geographic equatorial plane. That is, a rotation about the z-axis in Figure B1(a). Hence the geographic coordinates of the geomagnetic North pole are ( (90-) ^oN, ^oE ). The matrix, , is then given by:

We express the Cartesian components (x, y, z) of the general position vector, r, in terms of spherical polar coordinates

Substituting equation (B.2) for and equation (B.3) for r_m and r_g in equation (B.1) we can then solve for geomagnetic co-latitude and longitude in terms geographic co-latitude and longitude. We find:

and

For example, using = 11.5^o, = -69.0^o, the geographic coordinates of Halley (-75.5 ^oN, -26.6 ^oE) transform to the geomagnetic coordinates (-65.8 ^oN, 24.3 ^oE).