612 SCIENCE PROGRESS 



and \ X N = the number of growing cells reaching the dividing 



stage per second 



X 3 n = the number of dividing cells reaching the growing 



stage per second, 



dN 

 Then — =- = — \ X N + 2X 2 w 

 at 



Hence N = ^g[(« + *a)«"* - (£ + M* at ] 



and „ = ^%^- e -^l 

 a — p L J 



where a = - Xl + Xg + aA** + *»)' + 4*i*» 

 2 v 4 



__ Xi + X 2 a / ( x i + X z) 2 + 4^2 

 2 V 4 



These expressions reduce ultimately, after a few generations, 

 to the simpler expressions — 



... A^_ yg-^i _ i 



M Xi ./£ 



From the above £= (^i^) x i 



but o-/8 = -(Xi + X 8 ) 



. _X X 



A 



Since a satisfies the auxiliary equation of the differential 

 equation — 



a 2 + a(Xj + X 2 ) — X^ = O 



Inserting the value of \ 2 = ~ — a 



we obtain X 1 = a(2i£ + i) 



If we take the experimental fact that the volume of an 



average tumour of the rat sarcoma of Jensen doubles itself in 



four days, it is found that a = 2 x io* 6 sec" 1 . The further ex- 



N 

 penmental fact is that for such a tumour — = 100 approx. 



' n 



