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Proc. N. a. S- 
its own form of age distribution. For its constitution as to sex will in 
general differ from that of the "normal" population with self -perpetuating 
age distribution. We must therefore consider three different possibili- 
ties: 
1. The shaded area alone will produce a population exceeding at all 
ages the normal or fixed type continuation (8) of the original shaded area. 
Should this be the case, then the argument presented with regard to the 
case of invariable j8(a) holds a foriiori, so far as the minor tangent curve 
is concerned. 
2. The shaded area alone will produce a population deficient at some 
or all ages, as compared with the normal type continuation (8) of the origi- 
nal shaded area. In that case two alternatives present themselves: 
a. The deficiency is more than counterbalanced by the additional 
population produced by that portion of the original population which is 
represented by the dotted area in fig. 1. In this case also the original 
argument, so far as the minor tangent curve is concerned, applies essen- 
tially as before. 
b. After the contributions from all parts of the population have 
been taken into account, there remains an unbalanced deficiency short of 
the population defined by (8), in the population resulting from that origi- 
nally present. In such case the argument presented on the assumption 
of invariable ^(a) fails, and the population may move away from, not 
towards the age distribution (3). Stability of the fixed age distribution 
may not extend to such displacements as this. 
Similar reflections apply, mtitatis mutandis, as regards the major tangent 
curve. 
If conditions (1) or (2) prevail with respect to the minor, and corre- 
sponding conditions with respect to the major tangent curve, then we can 
argue as in the case of invariable i3(a), that after expiration of the existing 
generation a new pair of tangent curves can be drawn, which will lie within 
the pair defined by (8), (9). And if conditions (1) or (2) still persist with 
reference to the new tangent curves, the same process of closing in these 
curves at the expiration of the current generation can be repeated, and 
so on, as in the argument first presented. Now conditions (1) or (2) will 
thus continue to prevail for each new pair of tangents, if, as the minor and 
major tangents close up, j8(a) approaches a limiting form. In such case, 
therefore, the age distribution defined by (3), (4) is stable even for dis- 
placements of any magnitude, provided always that conditions (1) or (2) 
prevail, as indicated. 
1 Papers from the Department of Biometry and Vital Statistics, School of Hygiene 
and Public Health, Johns Hopkins University, No. 71. 
