22 



SCIENCE 



IN. S. Vol. XXVI. No. 653 



c(a) 





p(a) 



Now if general conditions in the commimity 

 are constant, c(a) will tend to assume a fixed 

 form. A little reflection shows that then hoth 

 N and B will increase in geometric progres- 

 sion with time,' at the same rate r= (6 — d). 

 We may, therefore, write: 



c{a)=J.e- 

 = lie-' 



"pia) 



(1) 



Now from the nature of the coefficient c(a) 

 it follows that 



C c{a]da=l 



Suhstituting this in (1) we have: 



1 z-** 



— = / e-''''p{a)da 

 b Jo 



(2) 



Equation (1) then gives the fixed age-dis- 

 tribution, while equation (2) (which may be 

 expanded into a series if desired), gives the 

 relation between h, the birth rate per head, 

 and r, the rate of natural increase per head, 

 and hence between h and d, since r=h — d. 



Applying these formulae to material fur- 

 nished by the Reports of the Registrar-General 

 of Births, etc., in England and Wales, the 

 following results were obtained: 



ENGLAND AND WALES 1871-80 (MEAN) 



Observed" Calculated 



Birth-rate per head 6 .03546 .0352 



Death-rate per head d .02139 .0211 



E.\oess .' {l} — d)=r .01407 (.0141) 



p{a) from Supplement to 45th Ann. Rep. Reg. 

 Gen. Births, etc., England and Wales, pp. vii and 

 viii, assuming ratio: 



male births 



female births 



= 1.04. 



^ Compare M. Block, " Traite thgorique et pra- 

 tique de statistique," 1886, p. 209. 



^ Jlean 6 and d from 46th Ann. Rep. Reg. Gen. 

 Births, etc., England and Wales, p. xxxi. 



Age Scale. — 1,000 individuals, in age-groups of 

 5 and 10 years 



It will be seen that in the above example 

 the values calculated for the age-scale and 

 especially for h and d, show a good agreement 

 with the observed values.' 



The above development admits of further 

 extension. But this, as well as further 

 numerical tests, must be reserved for a future 

 occasion. In view of the recent note of the 

 work by Major Woodruff, it appeared de- 

 sirable to the writer to publish this pre- 

 liminary note. 



Alfred J. Lotka 



A NEW GENUS AND SPECIES OP FOSSIL SHARK 

 RELATED fo EDESTUS LETOY 



The specimen which serves as the type of 

 the new genus and species, Lissoprion ferrieri, 

 was secured in what are regarded as Permo- 

 Carboniferous deposits near Montpelier, Bear 



Lake County, Idaho. It was collected by Mr. 

 W. F. Eerrier, of the town mentioned. The 

 specific name is given in his honor. The 



' The calculation is based on the observed value 

 of r=.0141, as indicated by the brackets. 



