Data for the Prol)lc,,t <>f Ecoli'tion 'm M.n. 163 



Series IV. EnyUxh Fatlwrs. 



(a) For all lives. Straight line : 



y = 2-4877 + 0-044,800* Origin of s, 



(b) For lives of 50 years and under. Straight line : j at birth and 



/ = -1-006 1+0-118,800.?; f unit of v one 



(c) For lives of 50 years and over. Straight line : year. 



y = 4-6717 + 0-014,400.^ 



(</) For all lives. Cubical parabola. Origin of ./ at 60 years and 

 unit = 5 years : 



y = 5-5075 + 0-153,403,'-- 0-041, 940.t' 2 + 0-003,636A 



The constants of the straight lines for all these series have been 

 found at once by fitting the best straight line to the observations, i.e., 

 by using the regression formula 



>j - mean tj = coefficient of regression ( or r~ } x (x - mean .*).* 



The cubical parabolas have been fitted by the method of moments. t 



The whole of this system of formulae has been plotted, and is 

 exhibited graphically in the accompanying diagrams (pp. 176 179). 

 These diagrams suffice to give the entire graphical solution of this prob- 

 lem to an exactness sufficient for most practical purposes. A careful ex- 

 amination of these diagrams will enable the reader to follow our general 

 conclusions even more clearly than inspection of the algebraic formulae. 



4. General Conclusions. (i.) The regression straight line for all lives, 

 ua, does not give a satisfactory picture of the relation between age at 

 death of a parent and the average number of offspring. We see at 

 once that it is too steep at the beginning and not steep enough at the 

 end of life. Accordingly, starting from 50 years as the sensible limit 

 to woman's child-bearing period, the mothers were broken up into two 

 groups, and the regression lines calculated separately for lives of 

 50 years and under, and for lives of 50 years and over. In this way 

 quite a reasonable fit was obtained to the observations. For con- 

 venience, the age of 50 was also taken as a dividing age for fathers. 

 In all four cases the regression line cc for parents living beyond 

 50 years shows a quite sensible deviation from the perpendicular, 

 or fertility is correlated with longevity even after the fecund period is 

 pasted. 



If we take American mothers there is no doubt of this increasing 



* See Yule, ' Roy. Soc. Proc.,' vol. CO, p. 477. 



t I hare shown in a memoir not yet published (a) how to fit all types of 

 curves, but particularly parabolas of any order, by the method of moments; and 

 (b) that sucli method gives results practically of the same order of exactness as 

 those given by the method of least squares. K. P. 



