It is pointed out that one of the formulas in Nambu's work on the analyticity of perturbation expansions can be simplified, and that as a consequence, the theory can beused to study the change of analyticity of a square diagram-the simplest four-pointdiagram-upon the insertion of internal lines in a not too complicated way. For N-,- scattering, it is shown that ( i ) insertion of any single internal line into the square diagram does not affect the original analyticity, (ii) insertion of any number of internal lines into the above square diagram, providedthat all are parallel to one of the sides (ladder approximation), does not affect the originalanalyticity, and (iii) insertion of two internal lines each connecting a pair of opposite sides affects the original analyticity, but the, resulting analyticity is still compatible with the, assumption in the theory of double dispersive relations.It is further shown that in a certain section in the nonphysical region, analyticity of any given diagram is not affected by the insertion of any number of internal lines. The effect of internal lines on anomalous thresholds is also discussed.