NO. 6 AGE OF THE EARTH — BECKER 13 



hausted. It seems worth while to attempt some estimates based on this 

 conception of the saltness of the ocean. 



There is a great deal of evidence for the elder Dana's generalization as 

 to the permanence of continental areas, and it is accepted by most physical 

 geologists. Assuming its truth it should be possible to arrive at a mean 

 value for the exposed land surface throughout geological time, and this 

 would be a constant of the same order of magnitude as the present con- 

 tinental area. 



The simplest law compatible with the conditions set forth is that, at any 

 given time, the decrease per unit of time in the area of the sodium-pro- 

 ducing exposures has been simply proportional to the temporary area of 

 the exposures. This is equivalent to the hypothesis that the area of the 

 feldspathic rocks can be represented approximately by the descending ex- 

 ponential ; for if y is the exposed area at a given time, t, and c a con- 

 stant, the decrease is represented by 



d,y/dt= —y/c, whence y = Ae~^^'' 



Here A is the extent of the exposure when ^ = 0, or when erosion is 

 supposed to begin. On the strength of Dana's law A may be taken as the 

 mean land area of the globe. 



Suppose the total sodium content of the ocean at time t to be N, and let 

 m be the annual yield of sodium per unit area, so that my is the increment 

 of N in one year from time t. Then, m being constant, 



N=\ mydt = Amc (l—y/A) 

 Jo 



or 



N/my 

 A/7-1 



while the formula for the age of the ocean is 



, 7 A 

 t — C lOOe 



y 



This hypothesis takes no account of a primitive saline ocean, though that 

 condition could be included by merely adding another constant. When t 

 becomes infinite, y reduces to zero and, therefore, the limiting value of 

 N is Amc. 



The selection of the exponential to represent the phenomena under 

 discussion is neither a random one nor dictated by mere convenience. 

 This function is well known to play a leading part in the theory of those 

 natural operations which may be classed generically as processes of absorp- 

 tion or gradual extinction," just as it also expresses the gradual accumula- 

 tion of money at compound interest. The descending exponential ex- 



1 Cf. Cournot, Theorie des fonctions et du calcul infinitesimal, vol. 1, chap. 2. 



