The various orders for the presentation of pairs in the method of paired comparisons are reviewed! It is shown that orders of the form worked out by Kowalewski for five, seven and fifteen members, if balanced to avoid time and space errors, are superior to any others. A general method for finding orders for any number of members is developed and the characteristics of the orders discussed. Methods are suggested for performing the calculation of the orders. A table of the balanced optimal orders for odd numbers of members from five to seventeen is given. The optimum, balanced orders have the following advantages: (1) They maintain the greatest possible spacing between pairs involving identical members, (2) they are so balanced as to remove time and space errors, (3) they avoid regular repetitions which might have suggestion effects, (4) by repeating the series in reverse order fatigue effects may be balanced out, (5) from these orders for odd-numbers of members, the optimum even-numbered orders may be obtained by a simple rule.

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