248 lotka: a natueal population noem 



the results shown in Tables II and III were found. These results 

 are also shown in graphic representation in figures 1, 2 and 3. 

 It will be seen that there is a remarkably clpee agreement between 

 observed and calculated values. 



In order to obtain an idea of the general character of the func- 

 tion defined by equations (2), (4) and Table I, the values of b 

 corresponding to a number of values of r were computed* and a 

 curve was plotted. The numbers so obtained are shown in Table 

 IV, and the curve in figure 4, It should be remarked that the 

 portion of the curve corresponding to high negative values of r 

 is of course only of geometrical interest — in nature such a value 

 could only occur under exceptional circumstances, and then only 

 for a limited time, as it would lead, in practise, to the extinction of 

 the species. 



* To be more precise, the computation was performed by the aid of another series 

 derived from (4). By (4) and Table I we have 



— = 41.35 - 1312r- + 29960r2 - SSOSOOr^ + 7.9 X 10 V - 9.99 X 10 V + 1.08 X 



lO^r^ - . . 

 This gives 



6m = 0.02418 + 0.7673r + 6.823r2 - 29.32r3 - 651.3r^ - . . . 

 When r = 0, 6 = 6o = rfo = 0.02418. Putting /i = 6 — 6o and reverting the 

 series by the method given by Prof. J. McMahon (Bull. Am. Math. Soc, April 

 1894, p. 170; see also C.'E. Van Orstrand, Phil. Mag., March, 1910, p. 366) we have: 



r = 1.3033/1 - 15. 10/^2 + ASiM^ - 12590/i* + 26500/i5 - . . . 

 and finally, since r = {b — d), 



d = 0.02418 - 0.3033/i + IS.lO/i^ - AMM^ + 12590/1^ - 265000/15 _|_ . . . 

 The actual computation was carried out by means of this last series. The rapidity 

 of the convergence of the series above is indicated by the number of terms given, 

 which is each case represents an accuracy of four significant figures in the result, 

 when r has a value of about 0.01400. 



