A. J. Lotka — Mode of Growth of Material Aggregates. 201 



The history of such an aggregate as we have been consider- 

 ing may be represented in a system of rectangular coordinates 

 by plotting as ordinates the values of N t c(a) corresponding to 

 the values of a measured along the Y axis and the values of t 

 measured along the X axis. The surface so obtained may be 

 called the N f c(a) surface ; any section of the same taken at 45° 

 to the planes of xz and yz is of the form : 



«'='N r .c(0)jp(a). 

 = B,„p(a), 



where t is the value of t corresponding to the point at which 

 the section cuts the X axis. 



Special Cases.. 

 A. Let c(a) be of fixed form. 



Then c(0) = ~ = constant = b say ; 



t 



" i'.i i , .-. D, p™ , . d lose p(a) , 

 while, by (4) -*f t = -J o c{a) a^A_Z. da 



= constant = d say ; 



d~N 

 hence —=— = B { — D^ = N t (b — d) — r~N say ; 



rt 



(10) 



Substituting these values in (3) we have : 



c(a) = be~ ra p(a). (11) 



b, d and r are respectively the rates "per head " of forma- 

 tion, of elimination, and of increase in number of individuals 

 in the aggregate. From the obvious relation, 



f 



c(a)da = 1 (12) 



or by (4) and (11), they must satisfy the condition : 



f 



-ra 



p{a)da 



'0 



Am. Jour. Sci.— Fourth Series, Vol. XXIV, No. 141.— September, 1907. 

 14 



