The Global Positioning System (GPS) transmits two frequencies, allowing users to correct for the first-order ionospheric signal group delay (or phase advance) of 1--50 m. The second-order ionospheric term, caused by the Faraday rotation effect induced by the Earth magnetic field, is about 1000 times smaller and usually ignored. In this study, we implement the 2nd-order correction suggested by Bassiri and Hajj <1993> and investigate its effect on GPS-inferred station positions. The correction causes a latitude dependent ~0.1--0.5 cm southward shift to the position which is roughly proportional to the integrated electron density above the receiver, and has strong diurnal, seasonal and decadal signatures. By analyzing a three-year time series of equatorial station positions obtained without the 2nd-order correction, a strong semi-annual north-south oscillation is observed, the origin of which has not been hitherto explained. We verify that this apparent oscillation can be largely removed once the 2nd-order correction is applied.