  It is well known that mutually orthogonal latin squares, or MOLS, admit a
(Kronecker) product construction. We show that, under mild conditions, `triple
products' of MOLS can result in a gain of one square. In terms of transversal
designs, the technique is to use a construction of Rolf Rees twice: once to
obtain a coarse resolution of the blocks after one product, and next to
reorganize classes and resolve the blocks of the second product. As
consequences, we report a few improvements to the MOLS table and obtain a
slight strengthening of the famous theorem of MacNeish.
