PRITCHETT — FORMULA FOR POPULATION. 



60 1 



A 



5 B -f 25 C — 125 D — 3.929 



A — 4 B + 16 C 



64 D 



5-308 



A — 3B-I- 9C — 27 D — 7-240 



A — 



A — 

 A 



A + 



A + 



2B 4- 

 B -h 



B + 

 2 B -f- 



4 C - 

 C — 



c -h 



4 C + 



8 D - 9.634 



D — 12.866 



— 17.069 



D — 23.192 



8 D — 31.443 



A + 3B-I- 9C+ 27 D- 38.558 

 A + 4B + 16C+ 64 D — 50.156 



= o 



= o 



= o 



= o 



= o 



= o 



== o 



= o 



= o 



= o 



V. 



+ 0.078 



— 0.038 



— 0.176 



— 0.060 



+ 0.II9 

 + 0.41 1 



+ 0.052 



— 0.9S2 



+ 0.758 



— 0.163 



Solving by the method of least squares, there result the follow- 

 ing normal equations : 



10 A — 



- 5 A + 

 + 85 A- 

 — 125 A + 



85 C — 125 D — 199.395 

 125 C + 1333 D — 307.645 

 125 B + 1333 C — 3125 D — 1598.197 

 1333 B — 3125 C + 25405 D — 3409.531 



5 B , 

 85 B — 

 125 B 



= o 



= o 



= o 



= o 



From their solution we obtain the most probable values of A, B, 

 C and D as follows : 



A = 4- 17-47969 



B = -f- 5.09SS0 

 C = -h 0.634506 

 D = -f- 0.0307275 



Accordingly, the population "P" for any time "z"' would be rep- 

 resented by the equation : — 



P = 17.47969 + 5.0988^ + 0.634506 t^ -f 0.0307275 i^ - - (i) 



This equation is evidently not what might be called a normal or 

 natural population curve. It has no asymptotes and P becomes 

 zero for a value of i equal to about —9.4, corresponding to the 

 year 1746. For larger negative values of/, P becomes negative. 

 This, however, is what is to be expected from the data used, since 

 the population there given is not the result of a slow natural 

 growth from an original small beginning, but is largely the result 

 of accretions from outside. 



