Gaussian Co-ordinates-background and derivation

[K29]

In Section XXV of Relativity we are introduced to Gaussian Co-ordinates. I can't seem to find much about this on the internet, perhaps there is another name? Can anyone link some background behind why this system was initially derived, and most importantly does anyone know the derivation of ds$$^{2}$$ =g11du$$^{2}$$+2g12dudv+g22dv$$^{2}$$ (formula for distance between two points on gaussian system)
Thanks in advance.

 

[HallsofIvy]

That's pretty much just saying that the distance between two points must be a

quadratic

since distance cannot be negative- and that's the most general possible quadratic formula in two dimensions.

 

[K29]

Thanks for that.