The map projection functions are X and Y, functions of s and t.
Thus the projection of r(s,t) onto the map is R(s, t) = [X(s,t), Y(s,t)]. Note that r is a 3 dimensional vector giving the location of the point in space. R is a 2 dimensional vector giving the location of the point on the map. So similarly to the equations (17) we can calculate what the two small displacements are.
So define...
As the 
's are small we have:-
Displacement 2 is similar.
Now for notational convenience, define...
Then the length of the displacement is :-
