304 Appendix 



Perks' law arises by using conditions (a), (b) and (c), 

 i.e. dfijdx = X(yi.-A) {E-tJi,)l{E-A) 



whence /x^, = ^ + ^ . -n .x. (^) 



The Perks (logistic) relation can be expressed as stating that the 

 rate of change of yi^ is proportional to the product of its value and the 

 amount by which it falls short of its upper limiting value. 



8. If the requirement of a constant upper limit for the rate of 

 mortality is relaxed other formulae can be developed on similar lines 

 to those of the preceding paragraph. For example, 



d^ _ X{fi^-A) _ 



dx - l+B(fji^-A) ^'^^' "^'^ ~^^ 



where Wj, = B{jx^—A) (5) 



and 



^ X{fM,-A)(l+^fi,-A 



dx ~ I 2D 



(l+-g/x,-^j 



gives /Lt, = ^ + ^(-l + \/l+4Z)e^^) (6) 



Formula (6) is equivalent to a continued fraction form for /x^, i.e. 



^■^^^ D?^ 



1 + ... 



and the relationship between formulae (2) to (6) is clearly seen by 

 expanding the expressions for /^^ in terms of powers of e^'^ i.e. 



formula (2) gives B e^' 

 „ (3) „ A+Be^' 



(4) „ A-\-B e^^'-BD e'^'^ + BD'' e^^ _. . . 



(5) „ A+Be^'-BDe''^ + lBD''e^^'^ —^' • 



(6) „ A+Be^-BDe''^ + 2BD^e^^' _. . . 



