484 PROFESSOR TAIT ON FORMULA REPRESENTING 



6. But F t can now be represented in a form almost as simple. For^ 



Ft=/t+/t+i+ +/ 49 



= * j(50-*)+ (49-*)+ + l} =hk(50-t) (51 -*) 



= \ k (50 — t) 2 , nearly enough for our purpose. 



7. Thus it appears that — 



Fecundity is proportional to the number of years a woman's age is under 



fifty ; and 

 Fertility at that age is proportional to the square of the same number. 



8. To show, numerically, how closely these formulae represent the tables is ot 

 course easy. 



Fecundity. 



Age 



15-19 



20-24 



25-29 



30-34 



35-39 



40-44 



45-49 



Dr Duncan 

 ^0 - 



50-0 

 49-5 



41-8 

 420 



346 

 345 



26-6 

 27-0 



20-4 

 195 



80 

 120 



1-8 



4-5 



Fertility. 



Age 



15-19 



20-24 



25-29 



30-34 



35-39 



40-44 



1 

 45-49 



Dr Duncan 



10-85 



8-24 



500 



400 





... 



... 



Calculated from/t 1 

 as in (4) J 



110 



816 



5-5 



3-4 



1-9 



0-55 



009 



10 V ' 



10-89 



784 



5-29 



3-24 



1-69 



0-64 



009 



9. Example. — As an application of the formula, let us suppose a woman, who 

 was married ten years ago at the age of twenty, to have now five children : — 



At marriage 

 At present 



: J h (50 - 20) 2 = 450 k 

 F 30 = i k (50 - 30) 2 = 200 k 



20 



But the difference F 20 — F 30 , or 250 k, represents five children. Hence F 30 , or 

 200 k, represents four more. So that her family will probably amount to nine. 



10. As illustrating the subject farther, I append portions of another of Dr 

 Duncan's tables {Trans. R. S. E., 1865-6, p. 306), with formulae for comparison 

 founded on the type f t = k (C — t). 



