Synopsis: According to the Chinese historiography of the Sungdynasty the courtiers YenSu and Wu Têjen constructed southpointing chariots of which brief descriptions exist. The longerstanding problem of reconstructing these chariots is attacked anew by a method which combines analyses of the text, engineering considerations and elementary algebraic geometry. The latter is used first to derive the requirements of geometry for successfully working southpointing chariots, and subsequently to show that the southpointing mechanism in YenSu's chariot must have been a special type of intermittently working gear assembly. The historical text combined with the requirements of enginering leads to a reconstruction in which an escapement lever synchronizes two gears.
The reconstruction of the Wu Têjen's chariot starts from the assumption  based on the symmetry of the assembly  that two differential gears were used. The gears may then be assembled in 15 different ways to construct southpointing mechanism. Using algebraic geometry again, the required trackwidth of the chariots incorporating these 15 gear assemblies are calculated and compared to the known width of the body of the chariot. In only three instances are the two widths approximately equal; of here three, one is to be preferred on grounds of adherence to the text and good engineering. The text is too brief that more than part of the mechanism which actuated jackwork in Wu Têjen's chariot can be reconstructed.
Without using elementary algebraic geometry it is impossible to limit sufficiently the number of possible reconstructions, but analysis of the texts, intuition and engineering knowledge are essential too. The results show that:
Mr. Lanchester's reconstruction cannot be valid as from the three horizontal gearwheels specified in the Sung Shih no differential gear train can be constructed. Furthermore each differential halves the angle from input to output, whereas the transmission of the orginal maintained 12 teeth from input (subordinate gear (B)) to output (big wheel (C)).
What can be learned from the text is:
So we nead a means to transmit the rotations of (B) to (C) intermittently. Gears (E ?) alone could not faciliate this, as each gear (D) not is mesh has somehow to store the rotational information until its duty cycle. Some translatorial movement of the axles of (D) is proposed. Further investigation shows, that gears (B) just have to move the centres of gears (D), not turning them ! The most probable construction is shown right. 
Now we have to build an alternator, which synchronizes the both wheels (D). The ominous 'crossbar' or 'transverse wood' comes in handy here: If set up as show left, it will have each wheel advance one tooth alternatingly as long as both want to. If one stops, the lever will stay at one side, thus unlocking the other (required if the chariot turns round one fixed point !) 
Now we put all this together. The function is described as follows:
Gears (D) are mounted on swivel arms pivoting round
shaft (V). Gears (B) pull these
arms forward, until the hook jumps off the tooth as depicted above (Some
sort of spring is needed to ths ends).

Mr. Lanchester's reconstruction again fails due to the lack of two central gears. But the availability of five gears at each side of the pole allows for different solutions:
Dimensions necessary for proper operation:
Just comparing the dimensions should convince everybody. 
A much simplier exercise in geometry is the construction of the turning 'boys' in the corners round the hsien, using the 13 equal wheels given by the text 
But one riddle still remains unsolved: What was the purpose of the 'back poles' and bamboo ropes ? As they are not used for the southpointing action (contrary to other reconstructions !), they may be part of some fancy motion works like waving the boys' hands on turning (some early indicators ?). Anyway, one possible way to meet the ancient description is given left. 