A method of constructing orthogonal arrays of an arbitrary strength $t$ is formulated. This method is a modification of the method based on differences, formulated by R. C. Bose [1] for the purpose of constructing orthogonal arrays of strength 2. It is shown further that each of the multifactorial designs of R. L. Plackett and J. P. Burman [2], in which each factor takes on two levels, provide a scheme for constructing orthogonal arrays of strength 3, consisting of the maximum possible number of rows. An orthogonal array (36, 13, 3, 2) is constructed. The method used for its construction cannot lead to a number of constraints greater than 13. It is known however [3] that 16 is an upper bound for the number of constraints in this case; the problem as to whether this bound can actually be attained remains unsolved.