We examine closely the motion of the south-pointing chariot on a surface by taking into account the fact that both wheels have to roll without slipping. We firstly develop a procedure of successive approximations. In the limit where the distance between the wheels tends to zero, we find again the result due to Santander that the pointer is parallel transported. This is longer true generally within the first-order approximation. We secondly determine in an exact manner the motion of the south-pointing chariot on the sphere. Then, we define a pointer and we prove that this one is parallel transported exactly.