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STATISTICS: PEARL AND REED 
Proc. N. a. S. 
paper, that is, location of the point of inflection in the middle and symmetry 
of the two limbs of the curve. Asymmetrical or skew curves of this sort 
can only arise when the equation, F\x) = 0, has no real roots. While 
any odd value of n may yield this form of curve the simplest equation 
that will do it is that in which n = 3, so that the equation of this curve 
becomes that of (vii) . 
Having determined that the growth within any one epoch or cycle may 
be approximately represented by equation (i), or more accurately by (vii), 
the next question is that of treating several epochs or cycles. Theoreti- 
cally, some form of (v) may be found by sufficient labor in the adjustment 
of constants so that one equation with say 5 or 7 constants would describe 
a long history of growth involving several cycles. Practically, however, 
we have found it easier and just as satisfactory in other respects to treat 
each cycle by itself. Since the cycles of any case of growth are additive, 
we may use for any single cycle the equation 
or more generally 
1 + me'"' ^ 
^ ^ 1 + me''''' + '''''' + ^^^^ 
where in both of these forms d represents the total growth attained in all the 
previous cycles. The term d is therefore the lower asmptote of the cycle 
of growth under consideration and d -\- kis its upper asymptote. 
In treating any two adjacent cycles, it should be noted that the lower 
asymptote of the second cycle is frequently below the upper asymptote 
of the first cycle, due to the fact that the second cycle is often started be- 
fore the first one has had time to reach its natural level. This for instance 
would be the case where a population entered upon an industrial era before 
the country had reached the limit of population possible under purely 
agricultural conditions. 
The theory presented in this paper has been found to be entirely suc- 
cessful in fitting the population growth of many different countries, and 
in a subsequent publication this fact will be demonstrated with examples. 
^ Papers from the Department of Biometry and Vital Statistics, School of Hygiene 
and Public Health, The Johns Hopkins University, No. 81. 
2 Pearl, R. and Reed, L. J., "On the Rate of Growth of the Population of the 
United States Since 1790 and Its Mathematical Representation." Proc. Nat. Acad. 
Sci., Vol. 6, pp. 275-288. 1920. 
