A square-root formulation, square-root time x excursion depth = k, developed at the Royal Navy Physiologic Laboratory (RNPL) predicts the safe duration of air excursions at 1 ATA for periods up to 6 h (air) and up to 3 h (heliox atmosphere). At pressures above 1 ATA, the value of k was initially considered to be proportional to Haldanian multiples of ATA and not to the square-root ATA used in this paper. Excursions of infinite (infinity) duration can also be predicted from a square-root formula derived from the systematic heliox saturation excursions conducted at the Experimental Diving Unit in which net excursion depth = k square-root P abs. This paper presents a broad spectrum format that extends the square-root principle: 1) to values of k above 1 ATA in the RNPL formula and 2) to excursions for (infinity) time from habitat depths to 820 ft; and 3) which demonstrates a relationship between fractional time excursions and excursions for infinite periods of time. Application of the generalization that net excursion depths are proportional to square-root P abs would reduce to a minimum the number of programmed test dives and would, in addition, allow such dives to be conducted at relatively low pressures, with only occasional validating tests at deep depths.