Constructing orthogonal latin square Parker/Knuth method
Find 10 transversals that don't intersect; these cover all the cells of the square.  Fill the transversal that starts with 0 will all 0's; fill the transversal that starts with 1 with all 1's, and so on.  In the example, the transversal that gives rise to the position of the 0's in the second square is 0349728615.  

It seems to me that it would be clearer to use the row numbers, rather than the number stored in that row, to indicate the transversal, but of course, I've used Euler's notation above.