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



The path A A' might, of course, also be of more compli- 

 cated form, having a number of maxima and minima. But such 

 a case does not seem to be of any particular interest in the 

 present connection. 



Let us now return to the case (7), as applied to the system 

 which we have been considering, viz., that in which the change 



A' 



is going on. 



A geometrical representation of the conditions in this system, 

 in accordance with the conception outlined above, ma} T be 

 obtained as follows : 



With our attention fixed first of all upon one particular 

 molecule, at a time when it is in the condition A', let a curve 

 be drawn, whose ordinates represent the values of E 7 , the free 

 energy of this molecule, corresponding to the times measured 

 off as abscissae. 



On the same diagram let a second curve be similarly drawn, 

 such that its ordinate for a given abscissa represents the mini- 

 mum value E of the free energy which the molecule would 

 have at that instant, supposing it to be in the state A, and to 

 retain its actual total energy. 



Lastly, on the same diagram, let a curve be drawn, whose 

 ordinate for a given abscissa represents the least maximum 

 value Ej, through which the free energy of the molecule must 

 pass under the existing conditions, if the molecule is at the 

 instant corresponding to the abscissa, transformed from the 

 state A 7 to the state A (limit of stability).* 



Now we have no means of directly ascertaining the exactf 

 form of these three curves for a given molecule, but we do 

 know something of the general aspect which they must 

 present. 



The principal features of interest are indicated in fig. 4, and 

 are best described by reference to this : 



We will suppose that, as shown in the figure, at the moment 

 t = 0, when we begin our observation, E 7 < E. As time goes 

 on E 7 and E vary somewhat, and the corresponding curves may 

 cross at some point t 1 at which E 7 = E. To the right of this 

 E 7 > E, and the first condition for the transformation A 7 A is 

 fulfilled. Following up these two curves further, we may 

 find them recrossing, but sooner or later we shall come to a 

 point where, with E < E 7 , the curve representing E 7 will meet 

 the third curve E r At this instant t„ the second condition 



* For the sake of simplicity it is here assumed that the path which con- 

 tains this least maximum, contains no other maximum. 



f The mean state of the molecule will presumably satisfy the law of 

 equipartition. See Rayleigh, Phil. Mag. (5), xlix, 98 ; Kelvin, Phil. Mag. (6), 

 ii, 1, 7; also W. F. Magie, Science, xxiii, 161, 1906. 



