Designing N:P (prime) Lego Technic Gear Ratios using two differentials

Here I present methods for designing N:P Lego Technic Gear ratios where N and P are prime using two differentials. Several differential topologies are presented and one cross connected topology is mathematically analysed and the relationship equation between the input and output derived. Using this equation and the computer solutions for the 1:N gear ratio designs, an example is worked through to create a 313 to 421 gear ratio. This gear ratio design is much more efficient than simply connecting an N:1 and 1:P design in series.