A Latin square of order n is an n-by-n array of n symbols, which we take to be the integers 0 to n-1, such that no symbol is repeated in any row or column. Two Latin squares of the same order are orthogonal if, when overlapped, no ordered pair of symbols occurs more than once. Equivalently, the Latin squares together form a bijection on the set of n-squared coordinates. In this thesis the question of what this bijection is in terms of projective planes is investigated. The major result here is a new necessary and sufficient condition such that two ternary rings correspond to the same plane.